Research ArticleA Stochastic Dynamic Programming ApproachBased on Bounded Rationality and Application toDynamic Portfolio Choice

Wenjie Bi Liuqing Tian Haiying Liu and Xiaohong Chen

Business School Central South University Changsha Hunan 410083 China

Correspondence should be addressed to Wenjie Bi biwenjiecsugmailcom

Received 14 March 2014 Accepted 5 May 2014 Published 22 May 2014

Academic Editor Fenghua Wen

Copyright copy 2014 Wenjie Bi et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Dynamic portfolio choice is an important problem in finance but the optimal strategy analysis is difficult when consideringmultiplestochastic volatility variables such as the stock price interest rate and income Besides recent research in experimental economicsindicates that the agent shows limited attention considering only the variables with high fluctuations but ignoring those with smallones By extending the sparse max method we propose an approach to solve dynamic programming problem with small stochasticvolatility and the agentrsquos bounded rationalityThis approach considers the agentrsquos behavioral factors and avoids effectively the ldquoCurseof Dimensionalityrdquo in a dynamic programming problem with more than a few state variables We then apply it to Merton dynamicportfolio choice model with stochastic volatility and get a tractable solution Finally the numerical analysis shows that the boundedrational agent may pay no attention to the varying equity premium and interest rate with small variance

1 Introduction

In reality how to choose an assetrsquos portfolio of consumptionand investment is one of the most important decisionsfor many people In modern portfolio choice field Merton[1 2] provides a general framework for understanding theportfolio demand of long-term investors when investmentopportunities change over time In a classical Merton model[1 2] however the riskless interest rate the risky mean rateof return and the volatility coefficient are usually assumedto be constant These assumptions are lack of realism partic-ularly over long time intervals A large volume of empiricalresearches in financial market which indicates the assump-tion that these variables are stochastic volatile and follow acertain stochastic process (eg Ornstein-Uhlenbeck process)is more realistic [3 4] But when introducing these stochasticvariables into the Merton-style portfolio choice model theproblem becomes increasingly complicated and formidableto solve Also this will lead to the ldquoCurse of DimensionalityrdquoQuite a lot of approaches have been developed to deal withthis kind of problems such as martingale methods [5ndash8] andvarious approximate numerical algorithms [9ndash12] Howeverthese methods have more restrictive assumptions and are too

complex to get a tractable solution of strong explanationsBased on the control of small noise Judd and Guu [13]proposed a method to solve dynamic programming prob-lems with stochastic disturbance He makes the simplifyingassumption that uncertainty is small and obtains the first- andhigh-order solutions of complicated dynamic programmingmodel This method provides a quite suitable solution fordynamic portfolio choice model with stochastic volatility

On the other hand a growing body of empirical studiesindicate that the agent considers only the variables withhigh fluctuations but ignores those with small ones [14ndash16]Bordalo et al [17] showed that the agent rationally chooses tobe inattentive to news Koszegi and Szeidl [18] analyzed themonetary policy and found out that when price is changedthe decision makers are usually unaware of it There are alsomany literatures showing that the agent pays attention tosalient factors Sims [19] uses two empirical strategies to ana-lyze how individuals optimize fully with respect to the incen-tives created by tax policies and shows that tax salience affectsagentsrsquo behavioral response Peng and Xiong [20] study theallocation of investorsrsquo attention among different informa-tion They find out that investors with limited attention willfocus on macroeconomic and industry information rather

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 840725 11 pageshttpdxdoiorg1011552014840725

2 Discrete Dynamics in Nature and Society

than that of a specific firm Seasholes and Wu [21] demon-strate that attention-grabbing events will attract investorsrsquoattention In their model they regard them as the proxyvariables and their results empirically indicate that theseevents have a significant impact on the allocation of investorrsquosattention Mackowiak and Wiederholt [22] show that deci-sion makersrsquo attention is usually drawn to salient payoffs

In recent years Gabaix [23] provides a sparse maxoperator to model dynamic programming with boundedrationality In the sparse max the agent pays less or no atten-tion to some features the fluctuations of which are smallerthan some thresholds and he tries to strike a good balancebetween the utility loss of inattention and the cognitive costwhich can be regarded as the loss for taking time to thinkabout the decisions rather than to enjoy oneself The sparsemax seems more realistic than traditional economic modelssince it has a very robust psychological foundation Alsoit can deal with problems of maximization with constraintseasily and get a tractable solution in a parsimonious way

However Gabaix [23] only studies the dynamic program-ming in a stationary environment without the stochasticvolatility terms But the financial market is strewn withnumerous stochastic dynamic programming problems andthese problems are hard to solve due to multitudinousstate variables To address this issue we extend the sparsemax operator and develop a stochastic version of Gabaixrsquosmethod The distinctive feature of this method is that itconsiders the agentrsquos behavioral factors (limited attention)and can effectively preclude the ldquoCurse ofDimensionalityrdquo formultiple variables To verify the validity and practicability ofour model we consider theMerton dynamic portfolio choiceproblemwith stochastic volatility variables (eg [24 25]) andget a tractable solution

The remainder of this paper is organized as followsSection 2 presents the sparse dynamic programming methodproposed by Gabaix [23] Section 3 extends this modeland gives a general principle for solving continuous-time dynamic programming with stochastic variables InSection 4 we apply our method toMerton dynamic portfoliochoice Finally we discuss some implications of our findingsand suggest topics for future research in Section 5

2 The Sparse Max Operator withoutConstraints

We mainly introduce the sparse max operator proposed byGabaix [23] in this section In the traditional version theagent faces a maximization problem

max119886

119906 (119886 y)

subject to 119887 (119886 y) ge 0(1)

where y = (1199101 1199102 119910

119899) 119906 is a utility function and 119887

is a constraint Variable 119886 and function 119887 have arbitrarydimensions For any optimal decision in principle thousandsof considerations are relevant to the agent Since it would betoo burdensome to take all of these variables into account theagent is used to discarding most of them At the same timehis attention is allocated purposefully to important variables

Hence the agent might sensibly pick a ldquosparserdquo represen-tation of the variables namely choose the attention vectorm = (119898

1 1198982 119898

119899) to replace variable 119910

119894with 119910

119904

119894=

119898119894119910119894119894 isin (1 2 119899) where the superscript 119904 of 119910119904

119894represents

sparse The optimal attention vector is obtained by weighingthe utility losses for imperfect inattention against the costsavings without thinking too much

The utility losses from imperfect inattention can beexpressed as follows [23]

119864 [V (m) minus V (l)] = minus12

sum

119894119895

(119898119894minus 1)Λ

119894119895(119898119895minus 1) + 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(2)

where V(m) = 119906(119886(y119904(m)) y) is the utility for a sparse agenty119904(m) = (119910

119904

1 119910119904

2 119910

119904

119899) l = (1 1 1)

119879 and V(l) is theutility when the agent is fully attentive 119900(y2) denotes thesecond-order infinitesimal of y Λ

119894119895= minus120590

1198941198951198862

119910119894

119906119886119886 where

120590119894119895= cov(119910

119894 119910119895) 119864(119910

119894) = 0 120590

119894is the standard deviation

of 119910119894 and 119886

119910119894= minus119906119886119886119906119886119910119894

which indicates by how much achange 119910

119894should change the action for traditional agent 119906

119886119886

is the second derivative of 119906 with respect to 119886 All derivativesabove are evaluated at y = 0 and the default action 119886

119889

=

argmax119886119906(119886 0)

Gabaix [23] assumes the cognitive cost is 119888(119898119894) = 120581|119898

119894|120573

where 120573 ge 0 and parameter 120581 ge 0 is a penalty for lack ofsparsity If 120581 = 0 the agent will be a traditional rational agent

Based on above analysis Gabaix [23] defines the sparsemax operator as follows

Definition 1 (see [23] Sparse max operator without con-straints) The sparsemax defined by the following procedure

Step 1 Choose the attention vectormlowast

mlowast = argminm

1

2

119899

sum

119894sdot119895=1

(1 minus 119898119894) Λ119894119895(1 minus 119898

119895) + 120581

119899

sum

119894=1

1003816100381610038161003816119898119894

1003816100381610038161003816

120573

(3)

Define 119910119904119894= 119898lowast

119894119910119894as the sparse representation of 119910

119894

Step 2 Choose the action

119886119904

= argmax119886

119906 (119886 y119904) (4)

and set the resulting utility to be 119906119904 = 119906(119886119904 y)Suppose y is one-dimensional vector formula (3) can be

transformed into119898lowast = min119898(12)(119898 minus 1)

2

1205902

+ 120581|119898| Gabaix[23] defines a function 119860 to represent the optimal attentionvector namely 119898lowast = 119860

120573(1205902

120581) = inf[argmin119898(12)(119898 minus

1)2

(1205902

120581) + |119898|120573

] and points out when 120573 = 1 the function119860120573(1205902

120581) satisfies the sparsity and continuity When 120573 = 1

and 120581 = 1 we have 1198601(1205902

) = max(1 minus 11205902 0) as shown inFigure 1 [23]

From Figure 1 we know that the agent will not considerthe variable when 0 le 1205902 le 120581 (120581 = 1)

When the vector y includes more than one variableand these variables perceived by the agent are uncorrelatedwe have 119898lowast

119894= 1198601(1205902

1198941198862

119910119894

|119906119886119886|120581) through formula (3) To

Discrete Dynamics in Nature and Society 3

0 1 2 3 4 5 60

02

04

06

08

1

12A1(120590

2)

1205902

Figure 1 The attention function 1198601

analyze the agentrsquos inattention expediently Gabaix [23]defines the truncation function 120591(119904 119901) = 119904max(1 minus 11990121199042 0)so we have 119898lowast

119894= 120591(1 120581120590

119886119886119910119894120590119894) Truncation function has

more intuitive economic implications a one-standard-deviation change of the variable 119910

119894makes the agent

change his action by 119886119910119894(120590119894120590119886) When 119886

119910119894(120590119894120590119886) is

small and satisfies |119886119910119894(120590119894120590119886)| le 120581 the agent will not

consider this factor Figure 2 shows the truncation function120591(119904 119901) = 119904max(1 minus 11990121199042 0) [23]

From Figure 2 we know that the agent who seeks ldquosparsi-tyrdquo should sensibly drop relatively unimportant features Inaddition if the features are larger than that cutoff they arestill dampened in Figure 2 120591(119904 119901) is below the 45 degree line(for positive 119904 in general |120591(119904 119901)| lt |119904|)

Based on the analysis above we can use the truncationfunction to represent the sparse agentrsquos optimal action

Remark 2 (see [23]) If rational optimal action is119886119903

(y) = 119886119889 +sum119894

119886119910119894119910119894+ 119900(

1003817100381710038171003817y1003817100381710038171003817)2

(119903 represents the rationality) (5)

which is obtained by the Taylor expansion around the defaultaction 119886119889 then the sparse agentrsquos optimal action is

119886119904

(y) = 119886119889 +sum119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

) (6)

where 120590119886is the standard deviation of 119886

3 A Stochastic Dynamic ProgrammingApproach Based on Sparse Max Operator

In order to effectively deal with stochastic dynamic program-ming in finance in this section we extend Gabaix [23] sparsemax operator and propose a bounded rational stochasticdynamic programming model in continuous time

The general model of stochastic dynamic programmingin continuous time is

maxintinfin

0

119890minus120588119905

119906 (119886 x y) 119889119905

119889x = 119892 (x (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (x (119905) 119886 (119905)) 119889119885x (119905)

119889y = ℎ (x (119905) y (119905)) 119889119905 + 120590 (y (119905)) 119889119861y (119905)

(7)

minus3 minus2 minus1 0 1 2 3

minus3

minus2

minus1

0

1

2

3

p

120591(s p)

Figure 2 The truncation function 120591(119904 119901) with 119901 = 1

where 120588 denotes the discount factor 119906 is the utility function119886 is the decision variable which has an arbitrary dimensionthe vector x represents important factors which are alwaysconsidered by the agent and the vector y defined in Section 2represents factors that that may not be considered by thesparse agent119892(x y 119886) ℎ(x y) are the state transition functionof x and y respectively And 120590(x 119886) 120590(y) represent thestochastic volatility of x and y respectively119889119885x119889119861y are inde-pendent standard Brownian motions namely 119889119885x119889119861y = 0We define the value function as119881(x y) = intinfin

0

119890minus120588119905

119906(119886 x y)119889119905

Assumption 3 Theutility function 119906 and value function119881 are119899-order continuously differentiable (119862119899 119899 ge 3)

Assumption 4 All state variables are stochastic and theyare independent of each other stochastic volatility of x isa function of x and 119886 while stochastic volatility of y isuncorrelated with 119886

Assumption 5 x is one dimensional that is only one variablewould be always considered by the agent and other variablesmay not be considered by the agent

Assumption 6 According to Judd and Guu [13] we assumethe variance of each component of vector y is small andindependent of one another

To facilitate analysis we use 119909 to replace x denote thestochastic differential equation of 119910

119894by 119889119910

119894= ℎ119894

(119909(119905)

119910119894(119905))119889119905 + 120590

119894

(119910119894(119905))119889119861

119910119894(119905) 119894 isin (1 2 119899) and use the

notation119863119909[119891] = 120597

119909[119891]+119886

119909120597119886[119891] as the total derivative with

respect to 119909 (ie the full impact of a change in 119909 includingthe impact it has on a change in the action 119886)

Based on Remark 2 in Section 2 we have the followingproposition

Proposition 7 The optimal action in bounded rationalitymodel (7) is

119886119904

(119909 y) = 119886119889 (119909) +sum119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

) (8)

where 120590119886is the standard deviation of 119886

4 Discrete Dynamics in Nature and Society

Proof See the appendix

From Proposition 7 we know that in order to derive theoptimal action 119886119904 we should get the default action 119886119889 whichis related to 119909 and 119886

119910119894 The detail process of solving them

is described as the following steps which contain the mainresults of our method

Step 1 Solve default action 119886119889By substituting y = 0 into the basic model (7) we get the

default model

maxintinfin

0

119890minus120588119905

119906 (119886 119909 0) 119889119905

119889119909 = 119892 (119909 (119905) 0 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

(9)

This is a general dynamic programmingmodel in continuoustime and the state variable is one dimension so we can getthe optimal default action 119886119889 and the value function119881(119909 0) =int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 easily [26]

Step 2 Solve 119886119910119894

The following Proposition 8 and its proof in the appendix

show the result and the process of obtaining 119886119910119894

Proposition 8 The impact of a change in 119910119894on the value

function is

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(10)

where

119881119910119894= (119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

119891 (119909) =

119881119909119909

119881

(11)

By implicit function theorem the impact of a changein 119910119894on the optimal action is 119886

119910119894= minus120601119910119894120601119886 where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(12)

Proof See the appendix

Now we can get the optimal action based on the twosteps above By the analysis of Proposition 7 we can seethat 120591(119886

119910119894 120581120590119886120590119894) represents the impact of variable 119910

119894on the

action 119886119904 When 119886119910119894is smaller than 120581120590

119886120590119894 120591(119886119910119894 120581120590119886120590119894) = 0

which means the agent will discard this factor

4 Application Dynamic Portfolio Choice

41 Merton Portfolio Problem with Stochastic Volatility Inthis section we consider a Merton dynamic portfolio choiceproblem with stochastic volatility in continuous time [24]In the traditional version of Merton model [1] the agentrsquosoptimal problem is

max 119864int

infin

0

119890minus120588119905

119906 (119888 (119905)) 119889119905

st 119889119908 (119905) = 119908 (119905) (119903 + (119887 minus 119903) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(13)

where 120588 is the discount factor 119906 is the utility function 119908(119905)is the wealth at time 119905 120590 is the standard deviation of 119908(119905)and 119888(119905) is the consumption at time 119905 The investment control120579(119905) at time 119905 is the fraction of the wealth invested in the riskyasset so (1minus120579(119905)) is the fraction of the wealth invested on theriskless asset 119903 is the riskless interest rate and 119887 is the riskymean rate of return 119889119882 follows standard Brownian motionWe assume the utility function 119906(119888) = 119888(119905)1minus120574(1 minus 120574) where120574 (0 lt 120574 lt 1) is the parameter of risk preference The goalis to choose consumption 119888(119905) and investment 120579(119905) controlprocesses to maximize long-term utility

In model (13) the riskless interest rate 119903 and the riskymean rate of return 119887 are assumed to be constant [1]Howeverthis assumption is unrealistic particularly over long timeintervals [27 28] Instead now we assume that these twovariables are stochastic and satisfying 119903(119905) = 119903 + 119903(119905) 119887(119905) =119887 +

119887(119905) where 119903 119887 represent the long mean of the riskless

interest rate and the risky rate of return respectively and119903(119905) 119887(119905) are their volatile part 119903(119905) and 119887(119905) depend on someldquoeconomic factorrdquo 120578(119905) [24] namely

119903 (119905) = 119877 (120578 (119905))

119887 (119905) = 119872 (120578 (119905))

(14)

And 120578(119905) satisfies the stochastic differential formula [24]

119889120578 = 119892 (120578 (119905)) 119889119905 + [120582119889119882 (119905) + (1 minus 1205822

)

12

119889 (119905)] (15)

where (119905) and 119882(119905) are independent standard Brownianmotions The parameter 120582 with |120582| lt 1 allows a correlationbetween theBrownianmotion119882(119905)driving the short rate andits volatility is the standard deviation of 120578

The budget equation of model (13) now becomes

119889119908 = 119908 (119905) (119903 (119905) + (119887 (119905) minus 119903 (119905)) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(16)

Discrete Dynamics in Nature and Society 5

We further assume that the stochastic differential equa-tion of 119903(119905) = 120578(119905) follows an Ornstein-Uhlenbeck process[25 29]

119889119903 = minus 120577119903(119903 minus 119903 (119905)) 119889119905

+ 120590119903[120582119903119889119882 (119905) + (1 minus 120582

2

119903)

12

119889119877 (119905)]

(17)

where 120577119903is the degree of mean reversion in expected excess

returns120590119903is the standard deviation of 119903(119905)119877(119905) is a Brownian

motion and independent of 119882(119905) and 120582119903is the coefficient

of 119908 and 119903 To simplify our model we assume 120582119903= 0

[30] By substituting 119903(119905) = 119903 + 119903(119905) into (17) we obtain119889(119903 + 119903(119905)) = minus120577

119903119903(119905)119889119905 + 120590

119903119889119877(119905) Since 119889119903 = 0 it follows

that 119889119903 = minus120577119903119903(119905)119889119905 + 120590

119903119889119877(119905)

We assume that 119887(119905) follows an Ornstein-Uhlenbeckprocess too so we obtain 119889119887 = minus120577

119887

119887(119905)119889119905 + 120590

119887119889119861(119905) where 120577

119887

has the same meaning with 120577119903 120590119887is the standard deviation

of 119887(119905) and we assume 120590119903and 120590

119887are small [13] 119861(119905) is a

Brownian motion and independent of 119877(119905) and 119882(119905) Thenwe get the following model

max 119864int

infin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905 (18)

st 119889119908 = 119908 (119905) ((119903 (119905) + 120579 (119905) (119887 (119905) minus 119903 (119905)))) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

119903 (119905) = 119903 + 119903 (119905)

119887 (119905) = 119887 +119887 (119905)

119889119903 = minus120577119903119903 (119905) 119889119905 + 120590

119903119889119877 (119905)

119889119887 = minus120577

119887

119887 (119905) 119889119905 + 120590

119887119889119861 (119905)

(19)

From Section 3 we know that for the bounded rationalagent the optimal consumption 119888119904 and the optimal fractionof wealth allocated to risky market 120579119904 in model (18) can beexpressed as

119888119904

= 119888119889

+ 120591(119888119903

120581120590119888

120590119903

) 119903 + 120591(119888119887

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(120579119903

120581120590120579

120590119903

) 119903 + 120591(120579119887

120581120590120579

120590119887

)119887

(20)

where 119888119889 and 120579119889 are the default actions when 119903(119905) = 0 119887(119905) =0 120590119888 120590120579are the standard deviation of 119888 and 120579 respectively

119888119903 119888119887 120579119903 and 120579

119887are the impact of 119903 and

119887 on 119888 and 120579respectively Next we will give the process of solving themusing the approach described in Section 3

Step 1 Solve the default decision 119888119889 and 120579119889By substituting 119903(119905) = 0 119887(119905) = 0 into the basic model

(18) we get the default model

max119864intinfin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905

119889119908 = 119908 (119905) (119903 + 120579 (119905) (119887 minus 119903)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(21)

This is the same Merton model as the model in (13) manyscholars have solved this problem [1 2] We define the valuefunction to be 119881(119908) = int

infin

0

119890minus120588119905

(119888(119905)1minus120574

(1 minus 120574))119889119905 then wehave [2]

119888119889

= 119908(R (1 minus 120574))minus1120574

120579119889

=

119887 minus 119903

1205741205902

119881 (119908) = R1199081minus120574

1 minus 120574

(22)

whereR = [(120588minus 119903(1minus 120574) minus (119887minus 119903)2

(1 minus 120574)21205902

120574)120574]minus120574

(1 minus 120574)

Step 2 Solve 119888119903 119888119887 120579119903 and 120579

119887

Next we will give the results of 119888119903 119888119887 120579119903 and 120579

119887

Proposition 9 shows their expressions and the proof is thesolution process

Proposition 9 Based on the results of (22) and implicitfunction theorem we have

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

120579119903= minus

1

1205741205902 120579

119887=

1

1205741205902

(23)

And the final results of model (18) are

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(24)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)2

(1 minus 120574)21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(25)

Proof See the appendix

Proposition 9 makes predictions about the sparse agentrsquoschoice When 120581 = 0 the agent is the traditional perfectlyrational agent And when 120581 gt 0 it is a policy of a sparseagent Larger 120581 indicates that the agent is less sensitive tofluctuations of both the riskless interest rate and the riskymean rate of return

6 Discrete Dynamics in Nature and Society

0 01 02 03 04 05 06 07 08 09 1minus22minus215minus21minus205

minus2minus195minus19minus185minus18

csr

120581

(a)

minus49minus488minus486minus484minus482minus48minus478minus476

csb

0 01 02 03 04 05 06 07 08 09 1 120581

(b)

Figure 3 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 15 120590

119887= 17

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

00501015020250303504

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 4 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 15 120590

119887= 17

42 Numerical Example Thepurpose of this numerical anal-ysis is to intuitively understand how the boundedly rationalagent changes its decisions with the changing of the variancesof factors Firstly we set 119888119904

119903= 120591(minus(1 minus 120579

119889

)119872 120581120590119888120590119903)

119888119904

119887

= 120591(minus1119873 120581120590119888120590119887) 120579119904119903= 120591(minus1120574120590

2

120581120590120579120590119903) and 120579119904

119887

=

120591(11205741205902

120581120590120579120590119887) Let 120574 = 06 120590 = 22 120590

119903= 15 120590

119887= 17 120590

120579=

05 120590119888= 13 119887 = 32 119903 = 18 120588 = 095 120577

119903= 02 120577

119887= 04 and

119908 = 5 thenwe have 119888119904119903 119888119904119887

as shown in Figures 3 and 4 In thesefigures the horizontal axis 120581 is an index of bounded rational-ity and 120581 isin [0 1] which is also applied to Figures 5 and 6

From Figure 3 we know that whatever 120581 is |119888119904119903| gt 0 and

|119888119904

119887

| gt 0 which means when the variances of 119903 and 119887 arebig the agent will consider them in the process of makinga decision

Figure 3(a) shows that if 120581 = 0 then the agent reacts likethe rational agent when 119903 goes up by 1 119888119904

119903will fall by minus219

(the agent saves more) For 120581 = 1 if 119903 goes up by 1 119888119904119903falls

by minus185 This result indicates that the greater the cognitivecost about the factor is the less attention will be paid to thisfactor by the boundedly rational agent From Figure 3(b) wecan reach a similar conclusion

Figure 4 also shows the agent will always consider 119903 and119887 that is |120579119904

119903| gt 0 and |120579119904

119887

| gt 0 whatever 120581 is It additionwe can obtain that if 120581 = 0 120579119904

119903= minus034 and 120579119904

119887

= 034whichmeans the rational agent has the same sensitivity aboutthe 119903 and 119887 when deciding 120579119904 With the increasing of 120581 theabsolute values of 120579119904

119903and 120579119904119887

both will decrease which means

that the agent will pay less attention to them In other wordsthe impact of 119903 and 119887 on 120579119904 will decrease for the increasingcognitive cost

Next we assume that the standard deviation of risklessinterest rate and the risky mean rate of return is smaller 120590

119903=

015 120590119887= 025 By keeping other parameters fixed we get

results shown in Figures 5 and 6Figure 5(a) shows thatwhen 120581 ge 026 119888119904

119903= 0whichmeans

that if the fluctuation is small the agent may discard 119903 whenhe decides the optimal consumption We can get a similarconclusion from Figure 5(b) when 120581 ge 095 119888119904

119903= 0 with

120590119887= 025From Figure 5(a) we know that if 120581 = 0 119888119904

119903= minus219

while Figure 3(a) also shows if 120581 = 0 119888119904119903= minus219 which

means for the rational agent the sensitivity of 119888119904 to 119903 hasnothing to do with 119903rsquos variance However the boundedlyrational agents have different reactions to 119903 as 120581 increasessuch as when 120581 = 026 119888119904

119903= 0 with 120590

119903= 015 in Figure 5(a)

while 119888119904119903= minus21 with 120590

119903= 15 in Figure 3(a) This disparity

indicates thatwhen the cognitive costs are the same and 120581 gt 0that is the agents have the same boundedly rational degreemore volatile factors will be considered while the factor withsmaller variance may be neglected

Additionally we can know that when 026 le 120581 lt 095the agent does not react to 119903 namely 119888119904

119903= 0 (in Figure 5(a))

but will react to a change in 119887 (in Figure 5(b)) which is moreimportant the sensitivity of 119888119904 to 119887 remains high even for ahigh cognitive friction 120581 Note that this ldquofeature by featurerdquo

Discrete Dynamics in Nature and Society 7

minus25

minus2

minus15

minus1

minus05

0

0 01 02 03 04 05 06 07 08 09 1 120581

csr

(a)

minus5minus45minus4

minus35minus3

minus25minus2

minus15minus1

minus050

0 01 02 03 04 05 06 07 08 09 1 120581

csb

(b)

Figure 5 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 015 120590

119887= 025

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

0005010150202503035

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 6 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 015 120590

119887= 025

selective attention could not be rationalized by just a fixedcost to consumption which is not feature dependent Butwhen 120581 ge 095 119888119904

119903= 119888119904

119887

= 0 which indicates that the agentwill pay no attention to both 119903 and 119887 once their thinking costsare beyond some thresholds

Considering Figure 6(a) we can see that when 120581 ge 011120579119904

119903= 0 while Figure 4(a) shows whatever 120581 is 120579119904

119903gt 0 with

120590119903= 15 which means the smaller the variance of a factor is

the more likely the agent will ignore it From Figure 6(b) wecan also obtain the same conclusion

5 Conclusion

Dynamic portfolio choice is an important but complexproblem in modern financial field but extant methodsalways generate complicated numerical calculations due tonumerous state variables Hence to address this problem thispaper extends the sparse max operator proposed by Gabaix[23] and proposes a new approach to deal with dynamicprogramming under stochastic terms under the assumptionof the agentrsquos limited attention We apply this method toMerton dynamic portfolio choice problem and find that iteffectively simplifies the modelrsquos solution process and avoidsthe ldquoCurse of Dimensionalityrdquo Finally numerical exampleshows that thismethod has significant economic implicationsand clearly interprets the agentrsquos economic behavior when hemakes a portfolio choice

Our study can be extended in several directions Futureresearch should consider the condition when the stochastic

factors are correlated with each other for it is more realisticBesides information faced by the agent is always impreciseand incomplete and the fuzzy set theory is an importantapproach to deal with this kind of problem [31ndash33] Henceusing fuzzy set theory to handle imprecise values in dynamicprogramming may be another direction for further research

Appendix

Proof of Proposition 7 Based on model (7) we define valuefunction

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119910119894119910119895

(A1)

For 119889119885119909119889119861y = 0 we have 120588119909119910119894 = 0 120588119910119894119910119895 = 0 where 120588119909119910119894 120588119910119894119910119895

represent coefficient between119909 and119910119894119910119894 and119910

119895 respectively

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

than that of a specific firm Seasholes and Wu [21] demon-strate that attention-grabbing events will attract investorsrsquoattention In their model they regard them as the proxyvariables and their results empirically indicate that theseevents have a significant impact on the allocation of investorrsquosattention Mackowiak and Wiederholt [22] show that deci-sion makersrsquo attention is usually drawn to salient payoffs

In recent years Gabaix [23] provides a sparse maxoperator to model dynamic programming with boundedrationality In the sparse max the agent pays less or no atten-tion to some features the fluctuations of which are smallerthan some thresholds and he tries to strike a good balancebetween the utility loss of inattention and the cognitive costwhich can be regarded as the loss for taking time to thinkabout the decisions rather than to enjoy oneself The sparsemax seems more realistic than traditional economic modelssince it has a very robust psychological foundation Alsoit can deal with problems of maximization with constraintseasily and get a tractable solution in a parsimonious way

However Gabaix [23] only studies the dynamic program-ming in a stationary environment without the stochasticvolatility terms But the financial market is strewn withnumerous stochastic dynamic programming problems andthese problems are hard to solve due to multitudinousstate variables To address this issue we extend the sparsemax operator and develop a stochastic version of Gabaixrsquosmethod The distinctive feature of this method is that itconsiders the agentrsquos behavioral factors (limited attention)and can effectively preclude the ldquoCurse ofDimensionalityrdquo formultiple variables To verify the validity and practicability ofour model we consider theMerton dynamic portfolio choiceproblemwith stochastic volatility variables (eg [24 25]) andget a tractable solution

The remainder of this paper is organized as followsSection 2 presents the sparse dynamic programming methodproposed by Gabaix [23] Section 3 extends this modeland gives a general principle for solving continuous-time dynamic programming with stochastic variables InSection 4 we apply our method toMerton dynamic portfoliochoice Finally we discuss some implications of our findingsand suggest topics for future research in Section 5

2 The Sparse Max Operator withoutConstraints

We mainly introduce the sparse max operator proposed byGabaix [23] in this section In the traditional version theagent faces a maximization problem

max119886

119906 (119886 y)

subject to 119887 (119886 y) ge 0(1)

where y = (1199101 1199102 119910

119899) 119906 is a utility function and 119887

is a constraint Variable 119886 and function 119887 have arbitrarydimensions For any optimal decision in principle thousandsof considerations are relevant to the agent Since it would betoo burdensome to take all of these variables into account theagent is used to discarding most of them At the same timehis attention is allocated purposefully to important variables

Hence the agent might sensibly pick a ldquosparserdquo represen-tation of the variables namely choose the attention vectorm = (119898

1 1198982 119898

119899) to replace variable 119910

119894with 119910

119904

119894=

119898119894119910119894119894 isin (1 2 119899) where the superscript 119904 of 119910119904

119894represents

sparse The optimal attention vector is obtained by weighingthe utility losses for imperfect inattention against the costsavings without thinking too much

The utility losses from imperfect inattention can beexpressed as follows [23]

119864 [V (m) minus V (l)] = minus12

sum

119894119895

(119898119894minus 1)Λ

119894119895(119898119895minus 1) + 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(2)

where V(m) = 119906(119886(y119904(m)) y) is the utility for a sparse agenty119904(m) = (119910

119904

1 119910119904

2 119910

119904

119899) l = (1 1 1)

119879 and V(l) is theutility when the agent is fully attentive 119900(y2) denotes thesecond-order infinitesimal of y Λ

119894119895= minus120590

1198941198951198862

119910119894

119906119886119886 where

120590119894119895= cov(119910

119894 119910119895) 119864(119910

119894) = 0 120590

119894is the standard deviation

of 119910119894 and 119886

119910119894= minus119906119886119886119906119886119910119894

which indicates by how much achange 119910

119894should change the action for traditional agent 119906

119886119886

is the second derivative of 119906 with respect to 119886 All derivativesabove are evaluated at y = 0 and the default action 119886

119889

=

argmax119886119906(119886 0)

Gabaix [23] assumes the cognitive cost is 119888(119898119894) = 120581|119898

119894|120573

where 120573 ge 0 and parameter 120581 ge 0 is a penalty for lack ofsparsity If 120581 = 0 the agent will be a traditional rational agent

Based on above analysis Gabaix [23] defines the sparsemax operator as follows

Definition 1 (see [23] Sparse max operator without con-straints) The sparsemax defined by the following procedure

Step 1 Choose the attention vectormlowast

mlowast = argminm

1

2

119899

sum

119894sdot119895=1

(1 minus 119898119894) Λ119894119895(1 minus 119898

119895) + 120581

119899

sum

119894=1

1003816100381610038161003816119898119894

1003816100381610038161003816

120573

(3)

Define 119910119904119894= 119898lowast

119894119910119894as the sparse representation of 119910

119894

Step 2 Choose the action

119886119904

= argmax119886

119906 (119886 y119904) (4)

and set the resulting utility to be 119906119904 = 119906(119886119904 y)Suppose y is one-dimensional vector formula (3) can be

transformed into119898lowast = min119898(12)(119898 minus 1)

2

1205902

+ 120581|119898| Gabaix[23] defines a function 119860 to represent the optimal attentionvector namely 119898lowast = 119860

120573(1205902

120581) = inf[argmin119898(12)(119898 minus

1)2

(1205902

120581) + |119898|120573

] and points out when 120573 = 1 the function119860120573(1205902

120581) satisfies the sparsity and continuity When 120573 = 1

and 120581 = 1 we have 1198601(1205902

) = max(1 minus 11205902 0) as shown inFigure 1 [23]

From Figure 1 we know that the agent will not considerthe variable when 0 le 1205902 le 120581 (120581 = 1)

When the vector y includes more than one variableand these variables perceived by the agent are uncorrelatedwe have 119898lowast

119894= 1198601(1205902

1198941198862

119910119894

|119906119886119886|120581) through formula (3) To

Discrete Dynamics in Nature and Society 3

0 1 2 3 4 5 60

02

04

06

08

1

12A1(120590

2)

1205902

Figure 1 The attention function 1198601

analyze the agentrsquos inattention expediently Gabaix [23]defines the truncation function 120591(119904 119901) = 119904max(1 minus 11990121199042 0)so we have 119898lowast

119894= 120591(1 120581120590

119886119886119910119894120590119894) Truncation function has

more intuitive economic implications a one-standard-deviation change of the variable 119910

119894makes the agent

change his action by 119886119910119894(120590119894120590119886) When 119886

119910119894(120590119894120590119886) is

small and satisfies |119886119910119894(120590119894120590119886)| le 120581 the agent will not

consider this factor Figure 2 shows the truncation function120591(119904 119901) = 119904max(1 minus 11990121199042 0) [23]

From Figure 2 we know that the agent who seeks ldquosparsi-tyrdquo should sensibly drop relatively unimportant features Inaddition if the features are larger than that cutoff they arestill dampened in Figure 2 120591(119904 119901) is below the 45 degree line(for positive 119904 in general |120591(119904 119901)| lt |119904|)

Based on the analysis above we can use the truncationfunction to represent the sparse agentrsquos optimal action

Remark 2 (see [23]) If rational optimal action is119886119903

(y) = 119886119889 +sum119894

119886119910119894119910119894+ 119900(

1003817100381710038171003817y1003817100381710038171003817)2

(119903 represents the rationality) (5)

which is obtained by the Taylor expansion around the defaultaction 119886119889 then the sparse agentrsquos optimal action is

119886119904

(y) = 119886119889 +sum119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

) (6)

where 120590119886is the standard deviation of 119886

3 A Stochastic Dynamic ProgrammingApproach Based on Sparse Max Operator

In order to effectively deal with stochastic dynamic program-ming in finance in this section we extend Gabaix [23] sparsemax operator and propose a bounded rational stochasticdynamic programming model in continuous time

The general model of stochastic dynamic programmingin continuous time is

maxintinfin

0

119890minus120588119905

119906 (119886 x y) 119889119905

119889x = 119892 (x (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (x (119905) 119886 (119905)) 119889119885x (119905)

119889y = ℎ (x (119905) y (119905)) 119889119905 + 120590 (y (119905)) 119889119861y (119905)

(7)

minus3 minus2 minus1 0 1 2 3

minus3

minus2

minus1

0

1

2

3

p

120591(s p)

Figure 2 The truncation function 120591(119904 119901) with 119901 = 1

where 120588 denotes the discount factor 119906 is the utility function119886 is the decision variable which has an arbitrary dimensionthe vector x represents important factors which are alwaysconsidered by the agent and the vector y defined in Section 2represents factors that that may not be considered by thesparse agent119892(x y 119886) ℎ(x y) are the state transition functionof x and y respectively And 120590(x 119886) 120590(y) represent thestochastic volatility of x and y respectively119889119885x119889119861y are inde-pendent standard Brownian motions namely 119889119885x119889119861y = 0We define the value function as119881(x y) = intinfin

0

119890minus120588119905

119906(119886 x y)119889119905

Assumption 3 Theutility function 119906 and value function119881 are119899-order continuously differentiable (119862119899 119899 ge 3)

Assumption 4 All state variables are stochastic and theyare independent of each other stochastic volatility of x isa function of x and 119886 while stochastic volatility of y isuncorrelated with 119886

Assumption 5 x is one dimensional that is only one variablewould be always considered by the agent and other variablesmay not be considered by the agent

Assumption 6 According to Judd and Guu [13] we assumethe variance of each component of vector y is small andindependent of one another

To facilitate analysis we use 119909 to replace x denote thestochastic differential equation of 119910

119894by 119889119910

119894= ℎ119894

(119909(119905)

119910119894(119905))119889119905 + 120590

119894

(119910119894(119905))119889119861

119910119894(119905) 119894 isin (1 2 119899) and use the

notation119863119909[119891] = 120597

119909[119891]+119886

119909120597119886[119891] as the total derivative with

respect to 119909 (ie the full impact of a change in 119909 includingthe impact it has on a change in the action 119886)

Based on Remark 2 in Section 2 we have the followingproposition

Proposition 7 The optimal action in bounded rationalitymodel (7) is

119886119904

(119909 y) = 119886119889 (119909) +sum119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

) (8)

where 120590119886is the standard deviation of 119886

4 Discrete Dynamics in Nature and Society

Proof See the appendix

From Proposition 7 we know that in order to derive theoptimal action 119886119904 we should get the default action 119886119889 whichis related to 119909 and 119886

119910119894 The detail process of solving them

is described as the following steps which contain the mainresults of our method

Step 1 Solve default action 119886119889By substituting y = 0 into the basic model (7) we get the

default model

maxintinfin

0

119890minus120588119905

119906 (119886 119909 0) 119889119905

119889119909 = 119892 (119909 (119905) 0 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

(9)

This is a general dynamic programmingmodel in continuoustime and the state variable is one dimension so we can getthe optimal default action 119886119889 and the value function119881(119909 0) =int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 easily [26]

Step 2 Solve 119886119910119894

The following Proposition 8 and its proof in the appendix

show the result and the process of obtaining 119886119910119894

Proposition 8 The impact of a change in 119910119894on the value

function is

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(10)

where

119881119910119894= (119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

119891 (119909) =

119881119909119909

119881

(11)

By implicit function theorem the impact of a changein 119910119894on the optimal action is 119886

119910119894= minus120601119910119894120601119886 where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(12)

Proof See the appendix

Now we can get the optimal action based on the twosteps above By the analysis of Proposition 7 we can seethat 120591(119886

119910119894 120581120590119886120590119894) represents the impact of variable 119910

119894on the

action 119886119904 When 119886119910119894is smaller than 120581120590

119886120590119894 120591(119886119910119894 120581120590119886120590119894) = 0

which means the agent will discard this factor

4 Application Dynamic Portfolio Choice

41 Merton Portfolio Problem with Stochastic Volatility Inthis section we consider a Merton dynamic portfolio choiceproblem with stochastic volatility in continuous time [24]In the traditional version of Merton model [1] the agentrsquosoptimal problem is

max 119864int

infin

0

119890minus120588119905

119906 (119888 (119905)) 119889119905

st 119889119908 (119905) = 119908 (119905) (119903 + (119887 minus 119903) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(13)

where 120588 is the discount factor 119906 is the utility function 119908(119905)is the wealth at time 119905 120590 is the standard deviation of 119908(119905)and 119888(119905) is the consumption at time 119905 The investment control120579(119905) at time 119905 is the fraction of the wealth invested in the riskyasset so (1minus120579(119905)) is the fraction of the wealth invested on theriskless asset 119903 is the riskless interest rate and 119887 is the riskymean rate of return 119889119882 follows standard Brownian motionWe assume the utility function 119906(119888) = 119888(119905)1minus120574(1 minus 120574) where120574 (0 lt 120574 lt 1) is the parameter of risk preference The goalis to choose consumption 119888(119905) and investment 120579(119905) controlprocesses to maximize long-term utility

In model (13) the riskless interest rate 119903 and the riskymean rate of return 119887 are assumed to be constant [1]Howeverthis assumption is unrealistic particularly over long timeintervals [27 28] Instead now we assume that these twovariables are stochastic and satisfying 119903(119905) = 119903 + 119903(119905) 119887(119905) =119887 +

119887(119905) where 119903 119887 represent the long mean of the riskless

interest rate and the risky rate of return respectively and119903(119905) 119887(119905) are their volatile part 119903(119905) and 119887(119905) depend on someldquoeconomic factorrdquo 120578(119905) [24] namely

119903 (119905) = 119877 (120578 (119905))

119887 (119905) = 119872 (120578 (119905))

(14)

And 120578(119905) satisfies the stochastic differential formula [24]

119889120578 = 119892 (120578 (119905)) 119889119905 + [120582119889119882 (119905) + (1 minus 1205822

)

12

119889 (119905)] (15)

where (119905) and 119882(119905) are independent standard Brownianmotions The parameter 120582 with |120582| lt 1 allows a correlationbetween theBrownianmotion119882(119905)driving the short rate andits volatility is the standard deviation of 120578

The budget equation of model (13) now becomes

119889119908 = 119908 (119905) (119903 (119905) + (119887 (119905) minus 119903 (119905)) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(16)

Discrete Dynamics in Nature and Society 5

We further assume that the stochastic differential equa-tion of 119903(119905) = 120578(119905) follows an Ornstein-Uhlenbeck process[25 29]

119889119903 = minus 120577119903(119903 minus 119903 (119905)) 119889119905

+ 120590119903[120582119903119889119882 (119905) + (1 minus 120582

2

119903)

12

119889119877 (119905)]

(17)

where 120577119903is the degree of mean reversion in expected excess

returns120590119903is the standard deviation of 119903(119905)119877(119905) is a Brownian

motion and independent of 119882(119905) and 120582119903is the coefficient

of 119908 and 119903 To simplify our model we assume 120582119903= 0

[30] By substituting 119903(119905) = 119903 + 119903(119905) into (17) we obtain119889(119903 + 119903(119905)) = minus120577

119903119903(119905)119889119905 + 120590

119903119889119877(119905) Since 119889119903 = 0 it follows

that 119889119903 = minus120577119903119903(119905)119889119905 + 120590

119903119889119877(119905)

We assume that 119887(119905) follows an Ornstein-Uhlenbeckprocess too so we obtain 119889119887 = minus120577

119887

119887(119905)119889119905 + 120590

119887119889119861(119905) where 120577

119887

has the same meaning with 120577119903 120590119887is the standard deviation

of 119887(119905) and we assume 120590119903and 120590

119887are small [13] 119861(119905) is a

Brownian motion and independent of 119877(119905) and 119882(119905) Thenwe get the following model

max 119864int

infin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905 (18)

st 119889119908 = 119908 (119905) ((119903 (119905) + 120579 (119905) (119887 (119905) minus 119903 (119905)))) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

119903 (119905) = 119903 + 119903 (119905)

119887 (119905) = 119887 +119887 (119905)

119889119903 = minus120577119903119903 (119905) 119889119905 + 120590

119903119889119877 (119905)

119889119887 = minus120577

119887

119887 (119905) 119889119905 + 120590

119887119889119861 (119905)

(19)

From Section 3 we know that for the bounded rationalagent the optimal consumption 119888119904 and the optimal fractionof wealth allocated to risky market 120579119904 in model (18) can beexpressed as

119888119904

= 119888119889

+ 120591(119888119903

120581120590119888

120590119903

) 119903 + 120591(119888119887

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(120579119903

120581120590120579

120590119903

) 119903 + 120591(120579119887

120581120590120579

120590119887

)119887

(20)

where 119888119889 and 120579119889 are the default actions when 119903(119905) = 0 119887(119905) =0 120590119888 120590120579are the standard deviation of 119888 and 120579 respectively

119888119903 119888119887 120579119903 and 120579

119887are the impact of 119903 and

119887 on 119888 and 120579respectively Next we will give the process of solving themusing the approach described in Section 3

Step 1 Solve the default decision 119888119889 and 120579119889By substituting 119903(119905) = 0 119887(119905) = 0 into the basic model

(18) we get the default model

max119864intinfin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905

119889119908 = 119908 (119905) (119903 + 120579 (119905) (119887 minus 119903)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(21)

This is the same Merton model as the model in (13) manyscholars have solved this problem [1 2] We define the valuefunction to be 119881(119908) = int

infin

0

119890minus120588119905

(119888(119905)1minus120574

(1 minus 120574))119889119905 then wehave [2]

119888119889

= 119908(R (1 minus 120574))minus1120574

120579119889

=

119887 minus 119903

1205741205902

119881 (119908) = R1199081minus120574

1 minus 120574

(22)

whereR = [(120588minus 119903(1minus 120574) minus (119887minus 119903)2

(1 minus 120574)21205902

120574)120574]minus120574

(1 minus 120574)

Step 2 Solve 119888119903 119888119887 120579119903 and 120579

119887

Next we will give the results of 119888119903 119888119887 120579119903 and 120579

119887

Proposition 9 shows their expressions and the proof is thesolution process

Proposition 9 Based on the results of (22) and implicitfunction theorem we have

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

120579119903= minus

1

1205741205902 120579

119887=

1

1205741205902

(23)

And the final results of model (18) are

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(24)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)2

(1 minus 120574)21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(25)

Proof See the appendix

Proposition 9 makes predictions about the sparse agentrsquoschoice When 120581 = 0 the agent is the traditional perfectlyrational agent And when 120581 gt 0 it is a policy of a sparseagent Larger 120581 indicates that the agent is less sensitive tofluctuations of both the riskless interest rate and the riskymean rate of return

6 Discrete Dynamics in Nature and Society

0 01 02 03 04 05 06 07 08 09 1minus22minus215minus21minus205

minus2minus195minus19minus185minus18

csr

120581

(a)

minus49minus488minus486minus484minus482minus48minus478minus476

csb

0 01 02 03 04 05 06 07 08 09 1 120581

(b)

Figure 3 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 15 120590

119887= 17

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

00501015020250303504

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 4 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 15 120590

119887= 17

42 Numerical Example Thepurpose of this numerical anal-ysis is to intuitively understand how the boundedly rationalagent changes its decisions with the changing of the variancesof factors Firstly we set 119888119904

119903= 120591(minus(1 minus 120579

119889

)119872 120581120590119888120590119903)

119888119904

119887

= 120591(minus1119873 120581120590119888120590119887) 120579119904119903= 120591(minus1120574120590

2

120581120590120579120590119903) and 120579119904

119887

=

120591(11205741205902

120581120590120579120590119887) Let 120574 = 06 120590 = 22 120590

119903= 15 120590

119887= 17 120590

120579=

05 120590119888= 13 119887 = 32 119903 = 18 120588 = 095 120577

119903= 02 120577

119887= 04 and

119908 = 5 thenwe have 119888119904119903 119888119904119887

as shown in Figures 3 and 4 In thesefigures the horizontal axis 120581 is an index of bounded rational-ity and 120581 isin [0 1] which is also applied to Figures 5 and 6

From Figure 3 we know that whatever 120581 is |119888119904119903| gt 0 and

|119888119904

119887

| gt 0 which means when the variances of 119903 and 119887 arebig the agent will consider them in the process of makinga decision

Figure 3(a) shows that if 120581 = 0 then the agent reacts likethe rational agent when 119903 goes up by 1 119888119904

119903will fall by minus219

(the agent saves more) For 120581 = 1 if 119903 goes up by 1 119888119904119903falls

by minus185 This result indicates that the greater the cognitivecost about the factor is the less attention will be paid to thisfactor by the boundedly rational agent From Figure 3(b) wecan reach a similar conclusion

Figure 4 also shows the agent will always consider 119903 and119887 that is |120579119904

119903| gt 0 and |120579119904

119887

| gt 0 whatever 120581 is It additionwe can obtain that if 120581 = 0 120579119904

119903= minus034 and 120579119904

119887

= 034whichmeans the rational agent has the same sensitivity aboutthe 119903 and 119887 when deciding 120579119904 With the increasing of 120581 theabsolute values of 120579119904

119903and 120579119904119887

both will decrease which means

that the agent will pay less attention to them In other wordsthe impact of 119903 and 119887 on 120579119904 will decrease for the increasingcognitive cost

Next we assume that the standard deviation of risklessinterest rate and the risky mean rate of return is smaller 120590

119903=

015 120590119887= 025 By keeping other parameters fixed we get

results shown in Figures 5 and 6Figure 5(a) shows thatwhen 120581 ge 026 119888119904

119903= 0whichmeans

that if the fluctuation is small the agent may discard 119903 whenhe decides the optimal consumption We can get a similarconclusion from Figure 5(b) when 120581 ge 095 119888119904

119903= 0 with

120590119887= 025From Figure 5(a) we know that if 120581 = 0 119888119904

119903= minus219

while Figure 3(a) also shows if 120581 = 0 119888119904119903= minus219 which

means for the rational agent the sensitivity of 119888119904 to 119903 hasnothing to do with 119903rsquos variance However the boundedlyrational agents have different reactions to 119903 as 120581 increasessuch as when 120581 = 026 119888119904

119903= 0 with 120590

119903= 015 in Figure 5(a)

while 119888119904119903= minus21 with 120590

119903= 15 in Figure 3(a) This disparity

indicates thatwhen the cognitive costs are the same and 120581 gt 0that is the agents have the same boundedly rational degreemore volatile factors will be considered while the factor withsmaller variance may be neglected

Additionally we can know that when 026 le 120581 lt 095the agent does not react to 119903 namely 119888119904

119903= 0 (in Figure 5(a))

but will react to a change in 119887 (in Figure 5(b)) which is moreimportant the sensitivity of 119888119904 to 119887 remains high even for ahigh cognitive friction 120581 Note that this ldquofeature by featurerdquo

Discrete Dynamics in Nature and Society 7

minus25

minus2

minus15

minus1

minus05

0

0 01 02 03 04 05 06 07 08 09 1 120581

csr

(a)

minus5minus45minus4

minus35minus3

minus25minus2

minus15minus1

minus050

0 01 02 03 04 05 06 07 08 09 1 120581

csb

(b)

Figure 5 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 015 120590

119887= 025

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

0005010150202503035

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 6 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 015 120590

119887= 025

selective attention could not be rationalized by just a fixedcost to consumption which is not feature dependent Butwhen 120581 ge 095 119888119904

119903= 119888119904

119887

= 0 which indicates that the agentwill pay no attention to both 119903 and 119887 once their thinking costsare beyond some thresholds

Considering Figure 6(a) we can see that when 120581 ge 011120579119904

119903= 0 while Figure 4(a) shows whatever 120581 is 120579119904

119903gt 0 with

120590119903= 15 which means the smaller the variance of a factor is

the more likely the agent will ignore it From Figure 6(b) wecan also obtain the same conclusion

5 Conclusion

Dynamic portfolio choice is an important but complexproblem in modern financial field but extant methodsalways generate complicated numerical calculations due tonumerous state variables Hence to address this problem thispaper extends the sparse max operator proposed by Gabaix[23] and proposes a new approach to deal with dynamicprogramming under stochastic terms under the assumptionof the agentrsquos limited attention We apply this method toMerton dynamic portfolio choice problem and find that iteffectively simplifies the modelrsquos solution process and avoidsthe ldquoCurse of Dimensionalityrdquo Finally numerical exampleshows that thismethod has significant economic implicationsand clearly interprets the agentrsquos economic behavior when hemakes a portfolio choice

Our study can be extended in several directions Futureresearch should consider the condition when the stochastic

factors are correlated with each other for it is more realisticBesides information faced by the agent is always impreciseand incomplete and the fuzzy set theory is an importantapproach to deal with this kind of problem [31ndash33] Henceusing fuzzy set theory to handle imprecise values in dynamicprogramming may be another direction for further research

Appendix

Proof of Proposition 7 Based on model (7) we define valuefunction

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119910119894119910119895

(A1)

For 119889119885119909119889119861y = 0 we have 120588119909119910119894 = 0 120588119910119894119910119895 = 0 where 120588119909119910119894 120588119910119894119910119895

represent coefficient between119909 and119910119894119910119894 and119910

119895 respectively

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

0 1 2 3 4 5 60

02

04

06

08

1

12A1(120590

2)

1205902

Figure 1 The attention function 1198601

analyze the agentrsquos inattention expediently Gabaix [23]defines the truncation function 120591(119904 119901) = 119904max(1 minus 11990121199042 0)so we have 119898lowast

119894= 120591(1 120581120590

119886119886119910119894120590119894) Truncation function has

more intuitive economic implications a one-standard-deviation change of the variable 119910

119894makes the agent

change his action by 119886119910119894(120590119894120590119886) When 119886

119910119894(120590119894120590119886) is

small and satisfies |119886119910119894(120590119894120590119886)| le 120581 the agent will not

consider this factor Figure 2 shows the truncation function120591(119904 119901) = 119904max(1 minus 11990121199042 0) [23]

From Figure 2 we know that the agent who seeks ldquosparsi-tyrdquo should sensibly drop relatively unimportant features Inaddition if the features are larger than that cutoff they arestill dampened in Figure 2 120591(119904 119901) is below the 45 degree line(for positive 119904 in general |120591(119904 119901)| lt |119904|)

Based on the analysis above we can use the truncationfunction to represent the sparse agentrsquos optimal action

Remark 2 (see [23]) If rational optimal action is119886119903

(y) = 119886119889 +sum119894

119886119910119894119910119894+ 119900(

1003817100381710038171003817y1003817100381710038171003817)2

(119903 represents the rationality) (5)

which is obtained by the Taylor expansion around the defaultaction 119886119889 then the sparse agentrsquos optimal action is

119886119904

(y) = 119886119889 +sum119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

) (6)

where 120590119886is the standard deviation of 119886

3 A Stochastic Dynamic ProgrammingApproach Based on Sparse Max Operator

In order to effectively deal with stochastic dynamic program-ming in finance in this section we extend Gabaix [23] sparsemax operator and propose a bounded rational stochasticdynamic programming model in continuous time

The general model of stochastic dynamic programmingin continuous time is

maxintinfin

0

119890minus120588119905

119906 (119886 x y) 119889119905

119889x = 119892 (x (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (x (119905) 119886 (119905)) 119889119885x (119905)

119889y = ℎ (x (119905) y (119905)) 119889119905 + 120590 (y (119905)) 119889119861y (119905)

(7)

minus3 minus2 minus1 0 1 2 3

minus3

minus2

minus1

0

1

2

3

p

120591(s p)

Figure 2 The truncation function 120591(119904 119901) with 119901 = 1

where 120588 denotes the discount factor 119906 is the utility function119886 is the decision variable which has an arbitrary dimensionthe vector x represents important factors which are alwaysconsidered by the agent and the vector y defined in Section 2represents factors that that may not be considered by thesparse agent119892(x y 119886) ℎ(x y) are the state transition functionof x and y respectively And 120590(x 119886) 120590(y) represent thestochastic volatility of x and y respectively119889119885x119889119861y are inde-pendent standard Brownian motions namely 119889119885x119889119861y = 0We define the value function as119881(x y) = intinfin

0

119890minus120588119905

119906(119886 x y)119889119905

Assumption 3 Theutility function 119906 and value function119881 are119899-order continuously differentiable (119862119899 119899 ge 3)

Assumption 4 All state variables are stochastic and theyare independent of each other stochastic volatility of x isa function of x and 119886 while stochastic volatility of y isuncorrelated with 119886

Assumption 5 x is one dimensional that is only one variablewould be always considered by the agent and other variablesmay not be considered by the agent

Assumption 6 According to Judd and Guu [13] we assumethe variance of each component of vector y is small andindependent of one another

To facilitate analysis we use 119909 to replace x denote thestochastic differential equation of 119910

119894by 119889119910

119894= ℎ119894

(119909(119905)

119910119894(119905))119889119905 + 120590

119894

(119910119894(119905))119889119861

119910119894(119905) 119894 isin (1 2 119899) and use the

notation119863119909[119891] = 120597

119909[119891]+119886

119909120597119886[119891] as the total derivative with

respect to 119909 (ie the full impact of a change in 119909 includingthe impact it has on a change in the action 119886)

Based on Remark 2 in Section 2 we have the followingproposition

Proposition 7 The optimal action in bounded rationalitymodel (7) is

119886119904

(119909 y) = 119886119889 (119909) +sum119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

) (8)

where 120590119886is the standard deviation of 119886

4 Discrete Dynamics in Nature and Society

Proof See the appendix

From Proposition 7 we know that in order to derive theoptimal action 119886119904 we should get the default action 119886119889 whichis related to 119909 and 119886

119910119894 The detail process of solving them

is described as the following steps which contain the mainresults of our method

Step 1 Solve default action 119886119889By substituting y = 0 into the basic model (7) we get the

default model

maxintinfin

0

119890minus120588119905

119906 (119886 119909 0) 119889119905

119889119909 = 119892 (119909 (119905) 0 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

(9)

This is a general dynamic programmingmodel in continuoustime and the state variable is one dimension so we can getthe optimal default action 119886119889 and the value function119881(119909 0) =int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 easily [26]

Step 2 Solve 119886119910119894

The following Proposition 8 and its proof in the appendix

show the result and the process of obtaining 119886119910119894

Proposition 8 The impact of a change in 119910119894on the value

function is

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(10)

where

119881119910119894= (119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

119891 (119909) =

119881119909119909

119881

(11)

By implicit function theorem the impact of a changein 119910119894on the optimal action is 119886

119910119894= minus120601119910119894120601119886 where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(12)

Proof See the appendix

Now we can get the optimal action based on the twosteps above By the analysis of Proposition 7 we can seethat 120591(119886

119910119894 120581120590119886120590119894) represents the impact of variable 119910

119894on the

action 119886119904 When 119886119910119894is smaller than 120581120590

119886120590119894 120591(119886119910119894 120581120590119886120590119894) = 0

which means the agent will discard this factor

4 Application Dynamic Portfolio Choice

41 Merton Portfolio Problem with Stochastic Volatility Inthis section we consider a Merton dynamic portfolio choiceproblem with stochastic volatility in continuous time [24]In the traditional version of Merton model [1] the agentrsquosoptimal problem is

max 119864int

infin

0

119890minus120588119905

119906 (119888 (119905)) 119889119905

st 119889119908 (119905) = 119908 (119905) (119903 + (119887 minus 119903) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(13)

where 120588 is the discount factor 119906 is the utility function 119908(119905)is the wealth at time 119905 120590 is the standard deviation of 119908(119905)and 119888(119905) is the consumption at time 119905 The investment control120579(119905) at time 119905 is the fraction of the wealth invested in the riskyasset so (1minus120579(119905)) is the fraction of the wealth invested on theriskless asset 119903 is the riskless interest rate and 119887 is the riskymean rate of return 119889119882 follows standard Brownian motionWe assume the utility function 119906(119888) = 119888(119905)1minus120574(1 minus 120574) where120574 (0 lt 120574 lt 1) is the parameter of risk preference The goalis to choose consumption 119888(119905) and investment 120579(119905) controlprocesses to maximize long-term utility

In model (13) the riskless interest rate 119903 and the riskymean rate of return 119887 are assumed to be constant [1]Howeverthis assumption is unrealistic particularly over long timeintervals [27 28] Instead now we assume that these twovariables are stochastic and satisfying 119903(119905) = 119903 + 119903(119905) 119887(119905) =119887 +

119887(119905) where 119903 119887 represent the long mean of the riskless

interest rate and the risky rate of return respectively and119903(119905) 119887(119905) are their volatile part 119903(119905) and 119887(119905) depend on someldquoeconomic factorrdquo 120578(119905) [24] namely

119903 (119905) = 119877 (120578 (119905))

119887 (119905) = 119872 (120578 (119905))

(14)

And 120578(119905) satisfies the stochastic differential formula [24]

119889120578 = 119892 (120578 (119905)) 119889119905 + [120582119889119882 (119905) + (1 minus 1205822

)

12

119889 (119905)] (15)

where (119905) and 119882(119905) are independent standard Brownianmotions The parameter 120582 with |120582| lt 1 allows a correlationbetween theBrownianmotion119882(119905)driving the short rate andits volatility is the standard deviation of 120578

The budget equation of model (13) now becomes

119889119908 = 119908 (119905) (119903 (119905) + (119887 (119905) minus 119903 (119905)) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(16)

Discrete Dynamics in Nature and Society 5

We further assume that the stochastic differential equa-tion of 119903(119905) = 120578(119905) follows an Ornstein-Uhlenbeck process[25 29]

119889119903 = minus 120577119903(119903 minus 119903 (119905)) 119889119905

+ 120590119903[120582119903119889119882 (119905) + (1 minus 120582

2

119903)

12

119889119877 (119905)]

(17)

where 120577119903is the degree of mean reversion in expected excess

returns120590119903is the standard deviation of 119903(119905)119877(119905) is a Brownian

motion and independent of 119882(119905) and 120582119903is the coefficient

of 119908 and 119903 To simplify our model we assume 120582119903= 0

[30] By substituting 119903(119905) = 119903 + 119903(119905) into (17) we obtain119889(119903 + 119903(119905)) = minus120577

119903119903(119905)119889119905 + 120590

119903119889119877(119905) Since 119889119903 = 0 it follows

that 119889119903 = minus120577119903119903(119905)119889119905 + 120590

119903119889119877(119905)

We assume that 119887(119905) follows an Ornstein-Uhlenbeckprocess too so we obtain 119889119887 = minus120577

119887

119887(119905)119889119905 + 120590

119887119889119861(119905) where 120577

119887

has the same meaning with 120577119903 120590119887is the standard deviation

of 119887(119905) and we assume 120590119903and 120590

119887are small [13] 119861(119905) is a

Brownian motion and independent of 119877(119905) and 119882(119905) Thenwe get the following model

max 119864int

infin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905 (18)

st 119889119908 = 119908 (119905) ((119903 (119905) + 120579 (119905) (119887 (119905) minus 119903 (119905)))) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

119903 (119905) = 119903 + 119903 (119905)

119887 (119905) = 119887 +119887 (119905)

119889119903 = minus120577119903119903 (119905) 119889119905 + 120590

119903119889119877 (119905)

119889119887 = minus120577

119887

119887 (119905) 119889119905 + 120590

119887119889119861 (119905)

(19)

From Section 3 we know that for the bounded rationalagent the optimal consumption 119888119904 and the optimal fractionof wealth allocated to risky market 120579119904 in model (18) can beexpressed as

119888119904

= 119888119889

+ 120591(119888119903

120581120590119888

120590119903

) 119903 + 120591(119888119887

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(120579119903

120581120590120579

120590119903

) 119903 + 120591(120579119887

120581120590120579

120590119887

)119887

(20)

where 119888119889 and 120579119889 are the default actions when 119903(119905) = 0 119887(119905) =0 120590119888 120590120579are the standard deviation of 119888 and 120579 respectively

119888119903 119888119887 120579119903 and 120579

119887are the impact of 119903 and

119887 on 119888 and 120579respectively Next we will give the process of solving themusing the approach described in Section 3

Step 1 Solve the default decision 119888119889 and 120579119889By substituting 119903(119905) = 0 119887(119905) = 0 into the basic model

(18) we get the default model

max119864intinfin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905

119889119908 = 119908 (119905) (119903 + 120579 (119905) (119887 minus 119903)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(21)

This is the same Merton model as the model in (13) manyscholars have solved this problem [1 2] We define the valuefunction to be 119881(119908) = int

infin

0

119890minus120588119905

(119888(119905)1minus120574

(1 minus 120574))119889119905 then wehave [2]

119888119889

= 119908(R (1 minus 120574))minus1120574

120579119889

=

119887 minus 119903

1205741205902

119881 (119908) = R1199081minus120574

1 minus 120574

(22)

whereR = [(120588minus 119903(1minus 120574) minus (119887minus 119903)2

(1 minus 120574)21205902

120574)120574]minus120574

(1 minus 120574)

Step 2 Solve 119888119903 119888119887 120579119903 and 120579

119887

Next we will give the results of 119888119903 119888119887 120579119903 and 120579

119887

Proposition 9 shows their expressions and the proof is thesolution process

Proposition 9 Based on the results of (22) and implicitfunction theorem we have

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

120579119903= minus

1

1205741205902 120579

119887=

1

1205741205902

(23)

And the final results of model (18) are

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(24)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)2

(1 minus 120574)21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(25)

Proof See the appendix

Proposition 9 makes predictions about the sparse agentrsquoschoice When 120581 = 0 the agent is the traditional perfectlyrational agent And when 120581 gt 0 it is a policy of a sparseagent Larger 120581 indicates that the agent is less sensitive tofluctuations of both the riskless interest rate and the riskymean rate of return

6 Discrete Dynamics in Nature and Society

0 01 02 03 04 05 06 07 08 09 1minus22minus215minus21minus205

minus2minus195minus19minus185minus18

csr

120581

(a)

minus49minus488minus486minus484minus482minus48minus478minus476

csb

0 01 02 03 04 05 06 07 08 09 1 120581

(b)

Figure 3 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 15 120590

119887= 17

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

00501015020250303504

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 4 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 15 120590

119887= 17

42 Numerical Example Thepurpose of this numerical anal-ysis is to intuitively understand how the boundedly rationalagent changes its decisions with the changing of the variancesof factors Firstly we set 119888119904

119903= 120591(minus(1 minus 120579

119889

)119872 120581120590119888120590119903)

119888119904

119887

= 120591(minus1119873 120581120590119888120590119887) 120579119904119903= 120591(minus1120574120590

2

120581120590120579120590119903) and 120579119904

119887

=

120591(11205741205902

120581120590120579120590119887) Let 120574 = 06 120590 = 22 120590

119903= 15 120590

119887= 17 120590

120579=

05 120590119888= 13 119887 = 32 119903 = 18 120588 = 095 120577

119903= 02 120577

119887= 04 and

119908 = 5 thenwe have 119888119904119903 119888119904119887

as shown in Figures 3 and 4 In thesefigures the horizontal axis 120581 is an index of bounded rational-ity and 120581 isin [0 1] which is also applied to Figures 5 and 6

From Figure 3 we know that whatever 120581 is |119888119904119903| gt 0 and

|119888119904

119887

| gt 0 which means when the variances of 119903 and 119887 arebig the agent will consider them in the process of makinga decision

Figure 3(a) shows that if 120581 = 0 then the agent reacts likethe rational agent when 119903 goes up by 1 119888119904

119903will fall by minus219

(the agent saves more) For 120581 = 1 if 119903 goes up by 1 119888119904119903falls

by minus185 This result indicates that the greater the cognitivecost about the factor is the less attention will be paid to thisfactor by the boundedly rational agent From Figure 3(b) wecan reach a similar conclusion

Figure 4 also shows the agent will always consider 119903 and119887 that is |120579119904

119903| gt 0 and |120579119904

119887

| gt 0 whatever 120581 is It additionwe can obtain that if 120581 = 0 120579119904

119903= minus034 and 120579119904

119887

= 034whichmeans the rational agent has the same sensitivity aboutthe 119903 and 119887 when deciding 120579119904 With the increasing of 120581 theabsolute values of 120579119904

119903and 120579119904119887

both will decrease which means

that the agent will pay less attention to them In other wordsthe impact of 119903 and 119887 on 120579119904 will decrease for the increasingcognitive cost

Next we assume that the standard deviation of risklessinterest rate and the risky mean rate of return is smaller 120590

119903=

015 120590119887= 025 By keeping other parameters fixed we get

results shown in Figures 5 and 6Figure 5(a) shows thatwhen 120581 ge 026 119888119904

119903= 0whichmeans

that if the fluctuation is small the agent may discard 119903 whenhe decides the optimal consumption We can get a similarconclusion from Figure 5(b) when 120581 ge 095 119888119904

119903= 0 with

120590119887= 025From Figure 5(a) we know that if 120581 = 0 119888119904

119903= minus219

while Figure 3(a) also shows if 120581 = 0 119888119904119903= minus219 which

means for the rational agent the sensitivity of 119888119904 to 119903 hasnothing to do with 119903rsquos variance However the boundedlyrational agents have different reactions to 119903 as 120581 increasessuch as when 120581 = 026 119888119904

119903= 0 with 120590

119903= 015 in Figure 5(a)

while 119888119904119903= minus21 with 120590

119903= 15 in Figure 3(a) This disparity

indicates thatwhen the cognitive costs are the same and 120581 gt 0that is the agents have the same boundedly rational degreemore volatile factors will be considered while the factor withsmaller variance may be neglected

Additionally we can know that when 026 le 120581 lt 095the agent does not react to 119903 namely 119888119904

119903= 0 (in Figure 5(a))

but will react to a change in 119887 (in Figure 5(b)) which is moreimportant the sensitivity of 119888119904 to 119887 remains high even for ahigh cognitive friction 120581 Note that this ldquofeature by featurerdquo

Discrete Dynamics in Nature and Society 7

minus25

minus2

minus15

minus1

minus05

0

0 01 02 03 04 05 06 07 08 09 1 120581

csr

(a)

minus5minus45minus4

minus35minus3

minus25minus2

minus15minus1

minus050

0 01 02 03 04 05 06 07 08 09 1 120581

csb

(b)

Figure 5 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 015 120590

119887= 025

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

0005010150202503035

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 6 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 015 120590

119887= 025

selective attention could not be rationalized by just a fixedcost to consumption which is not feature dependent Butwhen 120581 ge 095 119888119904

119903= 119888119904

119887

= 0 which indicates that the agentwill pay no attention to both 119903 and 119887 once their thinking costsare beyond some thresholds

Considering Figure 6(a) we can see that when 120581 ge 011120579119904

119903= 0 while Figure 4(a) shows whatever 120581 is 120579119904

119903gt 0 with

120590119903= 15 which means the smaller the variance of a factor is

the more likely the agent will ignore it From Figure 6(b) wecan also obtain the same conclusion

5 Conclusion

Dynamic portfolio choice is an important but complexproblem in modern financial field but extant methodsalways generate complicated numerical calculations due tonumerous state variables Hence to address this problem thispaper extends the sparse max operator proposed by Gabaix[23] and proposes a new approach to deal with dynamicprogramming under stochastic terms under the assumptionof the agentrsquos limited attention We apply this method toMerton dynamic portfolio choice problem and find that iteffectively simplifies the modelrsquos solution process and avoidsthe ldquoCurse of Dimensionalityrdquo Finally numerical exampleshows that thismethod has significant economic implicationsand clearly interprets the agentrsquos economic behavior when hemakes a portfolio choice

Our study can be extended in several directions Futureresearch should consider the condition when the stochastic

factors are correlated with each other for it is more realisticBesides information faced by the agent is always impreciseand incomplete and the fuzzy set theory is an importantapproach to deal with this kind of problem [31ndash33] Henceusing fuzzy set theory to handle imprecise values in dynamicprogramming may be another direction for further research

Appendix

Proof of Proposition 7 Based on model (7) we define valuefunction

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119910119894119910119895

(A1)

For 119889119885119909119889119861y = 0 we have 120588119909119910119894 = 0 120588119910119894119910119895 = 0 where 120588119909119910119894 120588119910119894119910119895

represent coefficient between119909 and119910119894119910119894 and119910

119895 respectively

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Proof See the appendix

From Proposition 7 we know that in order to derive theoptimal action 119886119904 we should get the default action 119886119889 whichis related to 119909 and 119886

119910119894 The detail process of solving them

is described as the following steps which contain the mainresults of our method

Step 1 Solve default action 119886119889By substituting y = 0 into the basic model (7) we get the

default model

maxintinfin

0

119890minus120588119905

119906 (119886 119909 0) 119889119905

119889119909 = 119892 (119909 (119905) 0 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

(9)

This is a general dynamic programmingmodel in continuoustime and the state variable is one dimension so we can getthe optimal default action 119886119889 and the value function119881(119909 0) =int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 easily [26]

Step 2 Solve 119886119910119894

The following Proposition 8 and its proof in the appendix

show the result and the process of obtaining 119886119910119894

Proposition 8 The impact of a change in 119910119894on the value

function is

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(10)

where

119881119910119894= (119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

119891 (119909) =

119881119909119909

119881

(11)

By implicit function theorem the impact of a changein 119910119894on the optimal action is 119886

119910119894= minus120601119910119894120601119886 where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(12)

Proof See the appendix

Now we can get the optimal action based on the twosteps above By the analysis of Proposition 7 we can seethat 120591(119886

119910119894 120581120590119886120590119894) represents the impact of variable 119910

119894on the

action 119886119904 When 119886119910119894is smaller than 120581120590

119886120590119894 120591(119886119910119894 120581120590119886120590119894) = 0

which means the agent will discard this factor

4 Application Dynamic Portfolio Choice

41 Merton Portfolio Problem with Stochastic Volatility Inthis section we consider a Merton dynamic portfolio choiceproblem with stochastic volatility in continuous time [24]In the traditional version of Merton model [1] the agentrsquosoptimal problem is

max 119864int

infin

0

119890minus120588119905

119906 (119888 (119905)) 119889119905

st 119889119908 (119905) = 119908 (119905) (119903 + (119887 minus 119903) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(13)

where 120588 is the discount factor 119906 is the utility function 119908(119905)is the wealth at time 119905 120590 is the standard deviation of 119908(119905)and 119888(119905) is the consumption at time 119905 The investment control120579(119905) at time 119905 is the fraction of the wealth invested in the riskyasset so (1minus120579(119905)) is the fraction of the wealth invested on theriskless asset 119903 is the riskless interest rate and 119887 is the riskymean rate of return 119889119882 follows standard Brownian motionWe assume the utility function 119906(119888) = 119888(119905)1minus120574(1 minus 120574) where120574 (0 lt 120574 lt 1) is the parameter of risk preference The goalis to choose consumption 119888(119905) and investment 120579(119905) controlprocesses to maximize long-term utility

In model (13) the riskless interest rate 119903 and the riskymean rate of return 119887 are assumed to be constant [1]Howeverthis assumption is unrealistic particularly over long timeintervals [27 28] Instead now we assume that these twovariables are stochastic and satisfying 119903(119905) = 119903 + 119903(119905) 119887(119905) =119887 +

119887(119905) where 119903 119887 represent the long mean of the riskless

interest rate and the risky rate of return respectively and119903(119905) 119887(119905) are their volatile part 119903(119905) and 119887(119905) depend on someldquoeconomic factorrdquo 120578(119905) [24] namely

119903 (119905) = 119877 (120578 (119905))

119887 (119905) = 119872 (120578 (119905))

(14)

And 120578(119905) satisfies the stochastic differential formula [24]

119889120578 = 119892 (120578 (119905)) 119889119905 + [120582119889119882 (119905) + (1 minus 1205822

)

12

119889 (119905)] (15)

where (119905) and 119882(119905) are independent standard Brownianmotions The parameter 120582 with |120582| lt 1 allows a correlationbetween theBrownianmotion119882(119905)driving the short rate andits volatility is the standard deviation of 120578

The budget equation of model (13) now becomes

119889119908 = 119908 (119905) (119903 (119905) + (119887 (119905) minus 119903 (119905)) 120579 (119905)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(16)

Discrete Dynamics in Nature and Society 5

We further assume that the stochastic differential equa-tion of 119903(119905) = 120578(119905) follows an Ornstein-Uhlenbeck process[25 29]

119889119903 = minus 120577119903(119903 minus 119903 (119905)) 119889119905

+ 120590119903[120582119903119889119882 (119905) + (1 minus 120582

2

119903)

12

119889119877 (119905)]

(17)

where 120577119903is the degree of mean reversion in expected excess

returns120590119903is the standard deviation of 119903(119905)119877(119905) is a Brownian

motion and independent of 119882(119905) and 120582119903is the coefficient

of 119908 and 119903 To simplify our model we assume 120582119903= 0

[30] By substituting 119903(119905) = 119903 + 119903(119905) into (17) we obtain119889(119903 + 119903(119905)) = minus120577

119903119903(119905)119889119905 + 120590

119903119889119877(119905) Since 119889119903 = 0 it follows

that 119889119903 = minus120577119903119903(119905)119889119905 + 120590

119903119889119877(119905)

We assume that 119887(119905) follows an Ornstein-Uhlenbeckprocess too so we obtain 119889119887 = minus120577

119887

119887(119905)119889119905 + 120590

119887119889119861(119905) where 120577

119887

has the same meaning with 120577119903 120590119887is the standard deviation

of 119887(119905) and we assume 120590119903and 120590

119887are small [13] 119861(119905) is a

Brownian motion and independent of 119877(119905) and 119882(119905) Thenwe get the following model

max 119864int

infin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905 (18)

st 119889119908 = 119908 (119905) ((119903 (119905) + 120579 (119905) (119887 (119905) minus 119903 (119905)))) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

119903 (119905) = 119903 + 119903 (119905)

119887 (119905) = 119887 +119887 (119905)

119889119903 = minus120577119903119903 (119905) 119889119905 + 120590

119903119889119877 (119905)

119889119887 = minus120577

119887

119887 (119905) 119889119905 + 120590

119887119889119861 (119905)

(19)

From Section 3 we know that for the bounded rationalagent the optimal consumption 119888119904 and the optimal fractionof wealth allocated to risky market 120579119904 in model (18) can beexpressed as

119888119904

= 119888119889

+ 120591(119888119903

120581120590119888

120590119903

) 119903 + 120591(119888119887

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(120579119903

120581120590120579

120590119903

) 119903 + 120591(120579119887

120581120590120579

120590119887

)119887

(20)

where 119888119889 and 120579119889 are the default actions when 119903(119905) = 0 119887(119905) =0 120590119888 120590120579are the standard deviation of 119888 and 120579 respectively

119888119903 119888119887 120579119903 and 120579

119887are the impact of 119903 and

119887 on 119888 and 120579respectively Next we will give the process of solving themusing the approach described in Section 3

Step 1 Solve the default decision 119888119889 and 120579119889By substituting 119903(119905) = 0 119887(119905) = 0 into the basic model

(18) we get the default model

max119864intinfin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905

119889119908 = 119908 (119905) (119903 + 120579 (119905) (119887 minus 119903)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(21)

This is the same Merton model as the model in (13) manyscholars have solved this problem [1 2] We define the valuefunction to be 119881(119908) = int

infin

0

119890minus120588119905

(119888(119905)1minus120574

(1 minus 120574))119889119905 then wehave [2]

119888119889

= 119908(R (1 minus 120574))minus1120574

120579119889

=

119887 minus 119903

1205741205902

119881 (119908) = R1199081minus120574

1 minus 120574

(22)

whereR = [(120588minus 119903(1minus 120574) minus (119887minus 119903)2

(1 minus 120574)21205902

120574)120574]minus120574

(1 minus 120574)

Step 2 Solve 119888119903 119888119887 120579119903 and 120579

119887

Next we will give the results of 119888119903 119888119887 120579119903 and 120579

119887

Proposition 9 shows their expressions and the proof is thesolution process

Proposition 9 Based on the results of (22) and implicitfunction theorem we have

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

120579119903= minus

1

1205741205902 120579

119887=

1

1205741205902

(23)

And the final results of model (18) are

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(24)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)2

(1 minus 120574)21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(25)

Proof See the appendix

Proposition 9 makes predictions about the sparse agentrsquoschoice When 120581 = 0 the agent is the traditional perfectlyrational agent And when 120581 gt 0 it is a policy of a sparseagent Larger 120581 indicates that the agent is less sensitive tofluctuations of both the riskless interest rate and the riskymean rate of return

6 Discrete Dynamics in Nature and Society

0 01 02 03 04 05 06 07 08 09 1minus22minus215minus21minus205

minus2minus195minus19minus185minus18

csr

120581

(a)

minus49minus488minus486minus484minus482minus48minus478minus476

csb

0 01 02 03 04 05 06 07 08 09 1 120581

(b)

Figure 3 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 15 120590

119887= 17

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

00501015020250303504

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 4 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 15 120590

119887= 17

42 Numerical Example Thepurpose of this numerical anal-ysis is to intuitively understand how the boundedly rationalagent changes its decisions with the changing of the variancesof factors Firstly we set 119888119904

119903= 120591(minus(1 minus 120579

119889

)119872 120581120590119888120590119903)

119888119904

119887

= 120591(minus1119873 120581120590119888120590119887) 120579119904119903= 120591(minus1120574120590

2

120581120590120579120590119903) and 120579119904

119887

=

120591(11205741205902

120581120590120579120590119887) Let 120574 = 06 120590 = 22 120590

119903= 15 120590

119887= 17 120590

120579=

05 120590119888= 13 119887 = 32 119903 = 18 120588 = 095 120577

119903= 02 120577

119887= 04 and

119908 = 5 thenwe have 119888119904119903 119888119904119887

as shown in Figures 3 and 4 In thesefigures the horizontal axis 120581 is an index of bounded rational-ity and 120581 isin [0 1] which is also applied to Figures 5 and 6

From Figure 3 we know that whatever 120581 is |119888119904119903| gt 0 and

|119888119904

119887

| gt 0 which means when the variances of 119903 and 119887 arebig the agent will consider them in the process of makinga decision

Figure 3(a) shows that if 120581 = 0 then the agent reacts likethe rational agent when 119903 goes up by 1 119888119904

119903will fall by minus219

(the agent saves more) For 120581 = 1 if 119903 goes up by 1 119888119904119903falls

by minus185 This result indicates that the greater the cognitivecost about the factor is the less attention will be paid to thisfactor by the boundedly rational agent From Figure 3(b) wecan reach a similar conclusion

Figure 4 also shows the agent will always consider 119903 and119887 that is |120579119904

119903| gt 0 and |120579119904

119887

| gt 0 whatever 120581 is It additionwe can obtain that if 120581 = 0 120579119904

119903= minus034 and 120579119904

119887

= 034whichmeans the rational agent has the same sensitivity aboutthe 119903 and 119887 when deciding 120579119904 With the increasing of 120581 theabsolute values of 120579119904

119903and 120579119904119887

both will decrease which means

that the agent will pay less attention to them In other wordsthe impact of 119903 and 119887 on 120579119904 will decrease for the increasingcognitive cost

Next we assume that the standard deviation of risklessinterest rate and the risky mean rate of return is smaller 120590

119903=

015 120590119887= 025 By keeping other parameters fixed we get

results shown in Figures 5 and 6Figure 5(a) shows thatwhen 120581 ge 026 119888119904

119903= 0whichmeans

that if the fluctuation is small the agent may discard 119903 whenhe decides the optimal consumption We can get a similarconclusion from Figure 5(b) when 120581 ge 095 119888119904

119903= 0 with

120590119887= 025From Figure 5(a) we know that if 120581 = 0 119888119904

119903= minus219

while Figure 3(a) also shows if 120581 = 0 119888119904119903= minus219 which

means for the rational agent the sensitivity of 119888119904 to 119903 hasnothing to do with 119903rsquos variance However the boundedlyrational agents have different reactions to 119903 as 120581 increasessuch as when 120581 = 026 119888119904

119903= 0 with 120590

119903= 015 in Figure 5(a)

while 119888119904119903= minus21 with 120590

119903= 15 in Figure 3(a) This disparity

indicates thatwhen the cognitive costs are the same and 120581 gt 0that is the agents have the same boundedly rational degreemore volatile factors will be considered while the factor withsmaller variance may be neglected

Additionally we can know that when 026 le 120581 lt 095the agent does not react to 119903 namely 119888119904

119903= 0 (in Figure 5(a))

but will react to a change in 119887 (in Figure 5(b)) which is moreimportant the sensitivity of 119888119904 to 119887 remains high even for ahigh cognitive friction 120581 Note that this ldquofeature by featurerdquo

Discrete Dynamics in Nature and Society 7

minus25

minus2

minus15

minus1

minus05

0

0 01 02 03 04 05 06 07 08 09 1 120581

csr

(a)

minus5minus45minus4

minus35minus3

minus25minus2

minus15minus1

minus050

0 01 02 03 04 05 06 07 08 09 1 120581

csb

(b)

Figure 5 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 015 120590

119887= 025

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

0005010150202503035

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 6 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 015 120590

119887= 025

selective attention could not be rationalized by just a fixedcost to consumption which is not feature dependent Butwhen 120581 ge 095 119888119904

119903= 119888119904

119887

= 0 which indicates that the agentwill pay no attention to both 119903 and 119887 once their thinking costsare beyond some thresholds

Considering Figure 6(a) we can see that when 120581 ge 011120579119904

119903= 0 while Figure 4(a) shows whatever 120581 is 120579119904

119903gt 0 with

120590119903= 15 which means the smaller the variance of a factor is

the more likely the agent will ignore it From Figure 6(b) wecan also obtain the same conclusion

5 Conclusion

Dynamic portfolio choice is an important but complexproblem in modern financial field but extant methodsalways generate complicated numerical calculations due tonumerous state variables Hence to address this problem thispaper extends the sparse max operator proposed by Gabaix[23] and proposes a new approach to deal with dynamicprogramming under stochastic terms under the assumptionof the agentrsquos limited attention We apply this method toMerton dynamic portfolio choice problem and find that iteffectively simplifies the modelrsquos solution process and avoidsthe ldquoCurse of Dimensionalityrdquo Finally numerical exampleshows that thismethod has significant economic implicationsand clearly interprets the agentrsquos economic behavior when hemakes a portfolio choice

Our study can be extended in several directions Futureresearch should consider the condition when the stochastic

factors are correlated with each other for it is more realisticBesides information faced by the agent is always impreciseand incomplete and the fuzzy set theory is an importantapproach to deal with this kind of problem [31ndash33] Henceusing fuzzy set theory to handle imprecise values in dynamicprogramming may be another direction for further research

Appendix

Proof of Proposition 7 Based on model (7) we define valuefunction

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119910119894119910119895

(A1)

For 119889119885119909119889119861y = 0 we have 120588119909119910119894 = 0 120588119910119894119910119895 = 0 where 120588119909119910119894 120588119910119894119910119895

represent coefficient between119909 and119910119894119910119894 and119910

119895 respectively

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

We further assume that the stochastic differential equa-tion of 119903(119905) = 120578(119905) follows an Ornstein-Uhlenbeck process[25 29]

119889119903 = minus 120577119903(119903 minus 119903 (119905)) 119889119905

+ 120590119903[120582119903119889119882 (119905) + (1 minus 120582

2

119903)

12

119889119877 (119905)]

(17)

where 120577119903is the degree of mean reversion in expected excess

returns120590119903is the standard deviation of 119903(119905)119877(119905) is a Brownian

motion and independent of 119882(119905) and 120582119903is the coefficient

of 119908 and 119903 To simplify our model we assume 120582119903= 0

[30] By substituting 119903(119905) = 119903 + 119903(119905) into (17) we obtain119889(119903 + 119903(119905)) = minus120577

119903119903(119905)119889119905 + 120590

119903119889119877(119905) Since 119889119903 = 0 it follows

that 119889119903 = minus120577119903119903(119905)119889119905 + 120590

119903119889119877(119905)

We assume that 119887(119905) follows an Ornstein-Uhlenbeckprocess too so we obtain 119889119887 = minus120577

119887

119887(119905)119889119905 + 120590

119887119889119861(119905) where 120577

119887

has the same meaning with 120577119903 120590119887is the standard deviation

of 119887(119905) and we assume 120590119903and 120590

119887are small [13] 119861(119905) is a

Brownian motion and independent of 119877(119905) and 119882(119905) Thenwe get the following model

max 119864int

infin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905 (18)

st 119889119908 = 119908 (119905) ((119903 (119905) + 120579 (119905) (119887 (119905) minus 119903 (119905)))) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

119903 (119905) = 119903 + 119903 (119905)

119887 (119905) = 119887 +119887 (119905)

119889119903 = minus120577119903119903 (119905) 119889119905 + 120590

119903119889119877 (119905)

119889119887 = minus120577

119887

119887 (119905) 119889119905 + 120590

119887119889119861 (119905)

(19)

From Section 3 we know that for the bounded rationalagent the optimal consumption 119888119904 and the optimal fractionof wealth allocated to risky market 120579119904 in model (18) can beexpressed as

119888119904

= 119888119889

+ 120591(119888119903

120581120590119888

120590119903

) 119903 + 120591(119888119887

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(120579119903

120581120590120579

120590119903

) 119903 + 120591(120579119887

120581120590120579

120590119887

)119887

(20)

where 119888119889 and 120579119889 are the default actions when 119903(119905) = 0 119887(119905) =0 120590119888 120590120579are the standard deviation of 119888 and 120579 respectively

119888119903 119888119887 120579119903 and 120579

119887are the impact of 119903 and

119887 on 119888 and 120579respectively Next we will give the process of solving themusing the approach described in Section 3

Step 1 Solve the default decision 119888119889 and 120579119889By substituting 119903(119905) = 0 119887(119905) = 0 into the basic model

(18) we get the default model

max119864intinfin

0

119890minus120588119905119888(119905)1minus120574

1 minus 120574

119889119905

119889119908 = 119908 (119905) (119903 + 120579 (119905) (119887 minus 119903)) 119889119905

+ 120579 (119905) 120590119908 (119905) 119889119882 (119905) minus 119888 (119905) 119889119905

(21)

This is the same Merton model as the model in (13) manyscholars have solved this problem [1 2] We define the valuefunction to be 119881(119908) = int

infin

0

119890minus120588119905

(119888(119905)1minus120574

(1 minus 120574))119889119905 then wehave [2]

119888119889

= 119908(R (1 minus 120574))minus1120574

120579119889

=

119887 minus 119903

1205741205902

119881 (119908) = R1199081minus120574

1 minus 120574

(22)

whereR = [(120588minus 119903(1minus 120574) minus (119887minus 119903)2

(1 minus 120574)21205902

120574)120574]minus120574

(1 minus 120574)

Step 2 Solve 119888119903 119888119887 120579119903 and 120579

119887

Next we will give the results of 119888119903 119888119887 120579119903 and 120579

119887

Proposition 9 shows their expressions and the proof is thesolution process

Proposition 9 Based on the results of (22) and implicitfunction theorem we have

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

120579119903= minus

1

1205741205902 120579

119887=

1

1205741205902

(23)

And the final results of model (18) are

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(24)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)2

(1 minus 120574)21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(25)

Proof See the appendix

Proposition 9 makes predictions about the sparse agentrsquoschoice When 120581 = 0 the agent is the traditional perfectlyrational agent And when 120581 gt 0 it is a policy of a sparseagent Larger 120581 indicates that the agent is less sensitive tofluctuations of both the riskless interest rate and the riskymean rate of return

6 Discrete Dynamics in Nature and Society

0 01 02 03 04 05 06 07 08 09 1minus22minus215minus21minus205

minus2minus195minus19minus185minus18

csr

120581

(a)

minus49minus488minus486minus484minus482minus48minus478minus476

csb

0 01 02 03 04 05 06 07 08 09 1 120581

(b)

Figure 3 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 15 120590

119887= 17

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

00501015020250303504

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 4 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 15 120590

119887= 17

42 Numerical Example Thepurpose of this numerical anal-ysis is to intuitively understand how the boundedly rationalagent changes its decisions with the changing of the variancesof factors Firstly we set 119888119904

119903= 120591(minus(1 minus 120579

119889

)119872 120581120590119888120590119903)

119888119904

119887

= 120591(minus1119873 120581120590119888120590119887) 120579119904119903= 120591(minus1120574120590

2

120581120590120579120590119903) and 120579119904

119887

=

120591(11205741205902

120581120590120579120590119887) Let 120574 = 06 120590 = 22 120590

119903= 15 120590

119887= 17 120590

120579=

05 120590119888= 13 119887 = 32 119903 = 18 120588 = 095 120577

119903= 02 120577

119887= 04 and

119908 = 5 thenwe have 119888119904119903 119888119904119887

as shown in Figures 3 and 4 In thesefigures the horizontal axis 120581 is an index of bounded rational-ity and 120581 isin [0 1] which is also applied to Figures 5 and 6

From Figure 3 we know that whatever 120581 is |119888119904119903| gt 0 and

|119888119904

119887

| gt 0 which means when the variances of 119903 and 119887 arebig the agent will consider them in the process of makinga decision

Figure 3(a) shows that if 120581 = 0 then the agent reacts likethe rational agent when 119903 goes up by 1 119888119904

119903will fall by minus219

(the agent saves more) For 120581 = 1 if 119903 goes up by 1 119888119904119903falls

by minus185 This result indicates that the greater the cognitivecost about the factor is the less attention will be paid to thisfactor by the boundedly rational agent From Figure 3(b) wecan reach a similar conclusion

Figure 4 also shows the agent will always consider 119903 and119887 that is |120579119904

119903| gt 0 and |120579119904

119887

| gt 0 whatever 120581 is It additionwe can obtain that if 120581 = 0 120579119904

119903= minus034 and 120579119904

119887

= 034whichmeans the rational agent has the same sensitivity aboutthe 119903 and 119887 when deciding 120579119904 With the increasing of 120581 theabsolute values of 120579119904

119903and 120579119904119887

both will decrease which means

that the agent will pay less attention to them In other wordsthe impact of 119903 and 119887 on 120579119904 will decrease for the increasingcognitive cost

Next we assume that the standard deviation of risklessinterest rate and the risky mean rate of return is smaller 120590

119903=

015 120590119887= 025 By keeping other parameters fixed we get

results shown in Figures 5 and 6Figure 5(a) shows thatwhen 120581 ge 026 119888119904

119903= 0whichmeans

that if the fluctuation is small the agent may discard 119903 whenhe decides the optimal consumption We can get a similarconclusion from Figure 5(b) when 120581 ge 095 119888119904

119903= 0 with

120590119887= 025From Figure 5(a) we know that if 120581 = 0 119888119904

119903= minus219

while Figure 3(a) also shows if 120581 = 0 119888119904119903= minus219 which

means for the rational agent the sensitivity of 119888119904 to 119903 hasnothing to do with 119903rsquos variance However the boundedlyrational agents have different reactions to 119903 as 120581 increasessuch as when 120581 = 026 119888119904

119903= 0 with 120590

119903= 015 in Figure 5(a)

while 119888119904119903= minus21 with 120590

119903= 15 in Figure 3(a) This disparity

indicates thatwhen the cognitive costs are the same and 120581 gt 0that is the agents have the same boundedly rational degreemore volatile factors will be considered while the factor withsmaller variance may be neglected

Additionally we can know that when 026 le 120581 lt 095the agent does not react to 119903 namely 119888119904

119903= 0 (in Figure 5(a))

but will react to a change in 119887 (in Figure 5(b)) which is moreimportant the sensitivity of 119888119904 to 119887 remains high even for ahigh cognitive friction 120581 Note that this ldquofeature by featurerdquo

Discrete Dynamics in Nature and Society 7

minus25

minus2

minus15

minus1

minus05

0

0 01 02 03 04 05 06 07 08 09 1 120581

csr

(a)

minus5minus45minus4

minus35minus3

minus25minus2

minus15minus1

minus050

0 01 02 03 04 05 06 07 08 09 1 120581

csb

(b)

Figure 5 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 015 120590

119887= 025

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

0005010150202503035

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 6 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 015 120590

119887= 025

selective attention could not be rationalized by just a fixedcost to consumption which is not feature dependent Butwhen 120581 ge 095 119888119904

119903= 119888119904

119887

= 0 which indicates that the agentwill pay no attention to both 119903 and 119887 once their thinking costsare beyond some thresholds

Considering Figure 6(a) we can see that when 120581 ge 011120579119904

119903= 0 while Figure 4(a) shows whatever 120581 is 120579119904

119903gt 0 with

120590119903= 15 which means the smaller the variance of a factor is

the more likely the agent will ignore it From Figure 6(b) wecan also obtain the same conclusion

5 Conclusion

Dynamic portfolio choice is an important but complexproblem in modern financial field but extant methodsalways generate complicated numerical calculations due tonumerous state variables Hence to address this problem thispaper extends the sparse max operator proposed by Gabaix[23] and proposes a new approach to deal with dynamicprogramming under stochastic terms under the assumptionof the agentrsquos limited attention We apply this method toMerton dynamic portfolio choice problem and find that iteffectively simplifies the modelrsquos solution process and avoidsthe ldquoCurse of Dimensionalityrdquo Finally numerical exampleshows that thismethod has significant economic implicationsand clearly interprets the agentrsquos economic behavior when hemakes a portfolio choice

Our study can be extended in several directions Futureresearch should consider the condition when the stochastic

factors are correlated with each other for it is more realisticBesides information faced by the agent is always impreciseand incomplete and the fuzzy set theory is an importantapproach to deal with this kind of problem [31ndash33] Henceusing fuzzy set theory to handle imprecise values in dynamicprogramming may be another direction for further research

Appendix

Proof of Proposition 7 Based on model (7) we define valuefunction

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119910119894119910119895

(A1)

For 119889119885119909119889119861y = 0 we have 120588119909119910119894 = 0 120588119910119894119910119895 = 0 where 120588119909119910119894 120588119910119894119910119895

represent coefficient between119909 and119910119894119910119894 and119910

119895 respectively

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

0 01 02 03 04 05 06 07 08 09 1minus22minus215minus21minus205

minus2minus195minus19minus185minus18

csr

120581

(a)

minus49minus488minus486minus484minus482minus48minus478minus476

csb

0 01 02 03 04 05 06 07 08 09 1 120581

(b)

Figure 3 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 15 120590

119887= 17

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

00501015020250303504

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 4 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 15 120590

119887= 17 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 15 120590

119887= 17

42 Numerical Example Thepurpose of this numerical anal-ysis is to intuitively understand how the boundedly rationalagent changes its decisions with the changing of the variancesof factors Firstly we set 119888119904

119903= 120591(minus(1 minus 120579

119889

)119872 120581120590119888120590119903)

119888119904

119887

= 120591(minus1119873 120581120590119888120590119887) 120579119904119903= 120591(minus1120574120590

2

120581120590120579120590119903) and 120579119904

119887

=

120591(11205741205902

120581120590120579120590119887) Let 120574 = 06 120590 = 22 120590

119903= 15 120590

119887= 17 120590

120579=

05 120590119888= 13 119887 = 32 119903 = 18 120588 = 095 120577

119903= 02 120577

119887= 04 and

119908 = 5 thenwe have 119888119904119903 119888119904119887

as shown in Figures 3 and 4 In thesefigures the horizontal axis 120581 is an index of bounded rational-ity and 120581 isin [0 1] which is also applied to Figures 5 and 6

From Figure 3 we know that whatever 120581 is |119888119904119903| gt 0 and

|119888119904

119887

| gt 0 which means when the variances of 119903 and 119887 arebig the agent will consider them in the process of makinga decision

Figure 3(a) shows that if 120581 = 0 then the agent reacts likethe rational agent when 119903 goes up by 1 119888119904

119903will fall by minus219

(the agent saves more) For 120581 = 1 if 119903 goes up by 1 119888119904119903falls

by minus185 This result indicates that the greater the cognitivecost about the factor is the less attention will be paid to thisfactor by the boundedly rational agent From Figure 3(b) wecan reach a similar conclusion

Figure 4 also shows the agent will always consider 119903 and119887 that is |120579119904

119903| gt 0 and |120579119904

119887

| gt 0 whatever 120581 is It additionwe can obtain that if 120581 = 0 120579119904

119903= minus034 and 120579119904

119887

= 034whichmeans the rational agent has the same sensitivity aboutthe 119903 and 119887 when deciding 120579119904 With the increasing of 120581 theabsolute values of 120579119904

119903and 120579119904119887

both will decrease which means

that the agent will pay less attention to them In other wordsthe impact of 119903 and 119887 on 120579119904 will decrease for the increasingcognitive cost

Next we assume that the standard deviation of risklessinterest rate and the risky mean rate of return is smaller 120590

119903=

015 120590119887= 025 By keeping other parameters fixed we get

results shown in Figures 5 and 6Figure 5(a) shows thatwhen 120581 ge 026 119888119904

119903= 0whichmeans

that if the fluctuation is small the agent may discard 119903 whenhe decides the optimal consumption We can get a similarconclusion from Figure 5(b) when 120581 ge 095 119888119904

119903= 0 with

120590119887= 025From Figure 5(a) we know that if 120581 = 0 119888119904

119903= minus219

while Figure 3(a) also shows if 120581 = 0 119888119904119903= minus219 which

means for the rational agent the sensitivity of 119888119904 to 119903 hasnothing to do with 119903rsquos variance However the boundedlyrational agents have different reactions to 119903 as 120581 increasessuch as when 120581 = 026 119888119904

119903= 0 with 120590

119903= 015 in Figure 5(a)

while 119888119904119903= minus21 with 120590

119903= 15 in Figure 3(a) This disparity

indicates thatwhen the cognitive costs are the same and 120581 gt 0that is the agents have the same boundedly rational degreemore volatile factors will be considered while the factor withsmaller variance may be neglected

Additionally we can know that when 026 le 120581 lt 095the agent does not react to 119903 namely 119888119904

119903= 0 (in Figure 5(a))

but will react to a change in 119887 (in Figure 5(b)) which is moreimportant the sensitivity of 119888119904 to 119887 remains high even for ahigh cognitive friction 120581 Note that this ldquofeature by featurerdquo

Discrete Dynamics in Nature and Society 7

minus25

minus2

minus15

minus1

minus05

0

0 01 02 03 04 05 06 07 08 09 1 120581

csr

(a)

minus5minus45minus4

minus35minus3

minus25minus2

minus15minus1

minus050

0 01 02 03 04 05 06 07 08 09 1 120581

csb

(b)

Figure 5 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 015 120590

119887= 025

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

0005010150202503035

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 6 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 015 120590

119887= 025

selective attention could not be rationalized by just a fixedcost to consumption which is not feature dependent Butwhen 120581 ge 095 119888119904

119903= 119888119904

119887

= 0 which indicates that the agentwill pay no attention to both 119903 and 119887 once their thinking costsare beyond some thresholds

Considering Figure 6(a) we can see that when 120581 ge 011120579119904

119903= 0 while Figure 4(a) shows whatever 120581 is 120579119904

119903gt 0 with

120590119903= 15 which means the smaller the variance of a factor is

the more likely the agent will ignore it From Figure 6(b) wecan also obtain the same conclusion

5 Conclusion

Dynamic portfolio choice is an important but complexproblem in modern financial field but extant methodsalways generate complicated numerical calculations due tonumerous state variables Hence to address this problem thispaper extends the sparse max operator proposed by Gabaix[23] and proposes a new approach to deal with dynamicprogramming under stochastic terms under the assumptionof the agentrsquos limited attention We apply this method toMerton dynamic portfolio choice problem and find that iteffectively simplifies the modelrsquos solution process and avoidsthe ldquoCurse of Dimensionalityrdquo Finally numerical exampleshows that thismethod has significant economic implicationsand clearly interprets the agentrsquos economic behavior when hemakes a portfolio choice

Our study can be extended in several directions Futureresearch should consider the condition when the stochastic

factors are correlated with each other for it is more realisticBesides information faced by the agent is always impreciseand incomplete and the fuzzy set theory is an importantapproach to deal with this kind of problem [31ndash33] Henceusing fuzzy set theory to handle imprecise values in dynamicprogramming may be another direction for further research

Appendix

Proof of Proposition 7 Based on model (7) we define valuefunction

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119910119894119910119895

(A1)

For 119889119885119909119889119861y = 0 we have 120588119909119910119894 = 0 120588119910119894119910119895 = 0 where 120588119909119910119894 120588119910119894119910119895

represent coefficient between119909 and119910119894119910119894 and119910

119895 respectively

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

minus25

minus2

minus15

minus1

minus05

0

0 01 02 03 04 05 06 07 08 09 1 120581

csr

(a)

minus5minus45minus4

minus35minus3

minus25minus2

minus15minus1

minus050

0 01 02 03 04 05 06 07 08 09 1 120581

csb

(b)

Figure 5 (a) Impact of a change in 119903 on 119888119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 119888119904 with 120590

119903= 015 120590

119887= 025

minus035minus03minus025minus02minus015minus01minus005

0

0 01 02 03 04 05 06 07 08 09 1 120581

120579sr

(a)

0005010150202503035

0 01 02 03 04 05 06 07 08 09 1 120581

120579sb

(b)

Figure 6 (a) Impact of a change in 119903 on 120579119904 with 120590119903= 015 120590

119887= 025 (b) Impact of a change in 119887 on 120579119904 with 120590

119903= 015 120590

119887= 025

selective attention could not be rationalized by just a fixedcost to consumption which is not feature dependent Butwhen 120581 ge 095 119888119904

119903= 119888119904

119887

= 0 which indicates that the agentwill pay no attention to both 119903 and 119887 once their thinking costsare beyond some thresholds

Considering Figure 6(a) we can see that when 120581 ge 011120579119904

119903= 0 while Figure 4(a) shows whatever 120581 is 120579119904

119903gt 0 with

120590119903= 15 which means the smaller the variance of a factor is

the more likely the agent will ignore it From Figure 6(b) wecan also obtain the same conclusion

5 Conclusion

Dynamic portfolio choice is an important but complexproblem in modern financial field but extant methodsalways generate complicated numerical calculations due tonumerous state variables Hence to address this problem thispaper extends the sparse max operator proposed by Gabaix[23] and proposes a new approach to deal with dynamicprogramming under stochastic terms under the assumptionof the agentrsquos limited attention We apply this method toMerton dynamic portfolio choice problem and find that iteffectively simplifies the modelrsquos solution process and avoidsthe ldquoCurse of Dimensionalityrdquo Finally numerical exampleshows that thismethod has significant economic implicationsand clearly interprets the agentrsquos economic behavior when hemakes a portfolio choice

Our study can be extended in several directions Futureresearch should consider the condition when the stochastic

factors are correlated with each other for it is more realisticBesides information faced by the agent is always impreciseand incomplete and the fuzzy set theory is an importantapproach to deal with this kind of problem [31ndash33] Henceusing fuzzy set theory to handle imprecise values in dynamicprogramming may be another direction for further research

Appendix

Proof of Proposition 7 Based on model (7) we define valuefunction

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119910119894119910119895

(A1)

For 119889119885119909119889119861y = 0 we have 120588119909119910119894 = 0 120588119910119894119910119895 = 0 where 120588119909119910119894 120588119910119894119910119895

represent coefficient between119909 and119910119894119910119894 and119910

119895 respectively

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Besides the volatility of a variable that may not be consideredby the agent is assumed to be small in Assumption 6 that is120590119894

(119910119894(119905)) = 0 so we have

119880lowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A2)

Similarly we define

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119904

119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119904

119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119894119895119881119904

119910119894119910119895

(A3)

From the analysis above we have

119880lowastlowast

(119886 119909 y) = 119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119904119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119904

119910119894

+

1

2

1205902

(119909 119886) 119881119904

119909119909

(A4)

Then the associated optimal actions can expressed as

119886lowast

(119909 y) = argmax119886

119880lowast

(119886 119909 y)

119886lowastlowast

(119909 y) = argmax119886

119880lowastlowast

(119886 119909 y) (A5)

First we will prove 120597119886lowast(119909 y)120597119910119894= 120597119886lowastlowast

(119909 y)120597119910119894at y =

0From the proof of lemma 1 in Gabaix [23] we know that

119881119904

(119909 y) = 119881(119909 y) + 119909120587(119909 y)119909 where 120587(119909 y) is continuous in(119909 y) and twice differentiable at y = 0 with 120587(119909 0) negativesemidefinite In other word the 119881119904(119909 y) and 119881

119904

(119909 y) =

119909120587(119909 y)119909 differ only by second-order terms in 119909 Thisbasically generalizes the envelope theorem It implies that at

y = 0 we have 119881119909(119909 0) = 119881

119904

119909(119909 0)

119881119909119909(119909 0) = 119881

119904

119909119909(119909 0)

119881119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

119881119909119910119894(119909 y)1003816100381610038161003816

1003816119910119894=0= 119881119904

119909119910119894

(119909 y)10038161003816100381610038161003816119910119894=0

(A6)

Differentiating formula (A2) with respect to 119886 gives

119880lowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A7)

Differentiating formula (A7) with respect to 119910119894and 119886

respectively we get

119880lowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886) 119881

119909119909

(A8)

Similarly differentiating formula (A4) with respect to 119886 gives

119880lowastlowast

119886= 119906119886+ 119892119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886(119909 119886) 119881

119904

119909119909 (A9)

Differentiating formula (A9) with respect to 119910119894and 119886

respectively gives

119880lowastlowast

119886119910119894

= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881119909

119904+ 119892119886(119909 y 119886) 119881119904

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119904

119909119910119894

]

119880lowast

119886119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881119904

119909+

1

2

1205902

119886119886(119909 119886) 119881

119904

119909119909

(A10)

Hence we have 119880lowast119886119910119894

= 119880lowastlowast

119886119910119894

at 119910119894= 0 and 119880lowast

119886119886= 119880lowastlowast

119886119886at

119910119894= 0 So

120597119886lowastlowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

= minus

119880lowastlowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowastlowast

119886119886

= minus

119880lowast

119886119910119894

10038161003816100381610038161003816119910119894=0

119880lowast

119886119886

=

120597119886lowast

(119909 y)120597119910119894

100381610038161003816100381610038161003816100381610038161003816119910119894=0

(A11)

Given 119886119903

(119909 y) = 119886119889

(119909) + sum119894119886119910119894119910119894+ 119900(y2) we

have 120597119886119903(119909 y)120597119910119894= 119886119910119894 According to (A11) we obtain

120597119886lowastlowast

(119909 y)120597119910119894= 119886119910119894 so 119886lowastlowast(119909 y) = 119886119889(119909)+sum

119894119886119910119894119910119894+119900(y2)

Finally

119886119904

(119909 y) = 119886 (119909mlowast119879y) = 119886119889 (119909)

+sum

119894

119886119910119894119898lowast

119894119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

1198861199101198941198601(

Λ119894119894

120581

) 119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

= 119886119889

(119909) +sum

119894

120591 (119886119910119894

120581120590119886

120590119894

)119910119894+ 119900 (

1003817100381710038171003817y1003817100381710038171003817

2

)

(A12)

where 120590119886is the standard deviation of 119886

Proof of Proposition 8 The laws of motion of model (7) are

119889119909 = 119892 (119909 (119905) y (119905) 119886 (119905)) 119889119905 + 120590 (119909 (119905) 119886 (119905)) 119889119885119909(119905)

119889119910119894= ℎ119894

(119909 (119905) 119910119894(119905)) 119889119905 + 120590

119894

(119910119894(119905)) 119889119861

119910119894(119905)

(A13)

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

where y = (1199101 1199102 119910

119899) Using Ito formula Bellmanrsquos

equation of model (7) can be expressed as follows

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

+

119899

sum

119894=1

1

2

(120590119894

(119910119894(119905)))

2

119881119910119894119910119894

+

119899

sum

119894=1

120590 (119909 119886) 120590119894

(119910119894(119905)) 120588119909119910119894119881119909119910119894

+

119899

sum

119894119895=1(119894 = 119895)

120590119894

(119910119894(119905)) 120590119895

(119910119895(119905)) 120588119910119894119910119895

119881119910119894119910119895

(A14)

From the proof of Proposition 7 above we have 120588119909119910119894

= 0120588119910119894119910119895

= 0 and 120590119894(119910119894(119905)) = 0 So we obtain

120588119881 = max119886

119906 (119886 119909 y) + 119892 (119909 y 119886) 119881119909

+

119899

sum

119894=1

ℎ119894

(119909 119910119894) 119881119910119894+

1

2

1205902

(119909 119886) 119881119909119909

(A15)

We define function 120601(119909 y 119886) as the derivative of the rightside in formula (A15) with respect to 119886 so 119886 satisfies 120601 = 0

with

120601 (119909 y 119886) = 119906119886+ 119892119886(119909 y 119886) 119881

119909+

1

2

1205902

119886(119909 119886) 119881

119909119909 (A16)

andwe define119891(119909) = 119881119909119909119881 where119891(119909) can be derived from

the expression of 119881(119909 0) = int

infin

0

119890minus120588119905

119906(119886 119909 0)119889119905 in Step 1 ofSection 3 Hence differentiating formula (A15) with respectto 119910119894gives

120588119881119910119894

= 119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909

+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

(119909 119886) 119891 (119909)119881119910119894

(A17)

Now we differentiate at y = 0 and evaluate at (119886 119909 y) =(119886119889

119909 0)

120588119881119909119910119894

= 119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) 119881119909119910119894

+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) (119891119909(119909) 119881119910119894+ 119891 (119909)119881

119909119910119894)

(A18)

From (A17) we get 119881119910119894

= (119906119886119910119894

+ 119892119910119894(119909 y 119886)119881

119909)(120588 minus

sum119899

119894=1ℎ119894

119910119894

(119909 119910119894 119886) minus (12)120590

2

(119909 119886)119891(119909))

And from (A18) we obtain

119881119909119910119894

= (119863119909[119906119886119910119894+ 119892119910119894(119909 y 119886) 119881

119909]

+

119899

sum

119894=1

ℎ119894

119909119910119894

(119909 119910119894 119886) 119881119910119894+

1

2

1205902

119909(119909 119886) 119891 (119909)119881

119910119894

+

1

2

1205902

(119909 119886) 119891119909(119909)119881119910119894)

times (120588 minus

119899

sum

119894=1

ℎ119894

119910119894

(119909 119910119894 119886) minus

1

2

1205902

(119909 119886) 119891 (119909))

minus1

(A19)

According to formula (A16) we know that the impact of119910119894on the optimal action can be expressed as 119886

119910119894= minus120601119910119894120601119886

where

120601119886= 119906119886119886+ 119892119886119886(119909 y 119886) 119881

119909+

1

2

1205902

119886119886(119909 119886)119863

119909[119881119909]

120601119910119894= 119906119886119910119894+ 119892119886119910119894(119909 y 119886) 119881

119909+ 119892119886(119909 y 119886) 119881

119909119910119894

+

1

2

1205902

119886(119909 119886)119863

119909[119881119909119910119894]

(A20)

Proof of Proposition 9 Using Ito formula Bellmanrsquos formulaof model (18) is

120588119881 = max119888

119906 (119888)

+ [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887

+

1

2

(120579120590119908)2

119881119908119908

+

1

2

1205902

119903119881119903119903+

1

2

1205902

119887119881119887119887

+ 120590120590119903120588119908119903119881119908119903+ 120590120590119887120588119908119887119881119908119887+ 120590119903120590119887120588119903119887119881119903119887

(A21)

where 120588119908119903 120588119908119887 and 120588

119887119903represent the coefficient between 119908

and 119903 119908 and 119887 and 119887 and 119903 respectively For 119889119882119889119877 = 0119889119882119889119861 = 0 and 119889119877119889119861 = 0 we have 120588

119908119903= 0 120588

119908119887= 0 and

120588119887119903= 0 According to Assumption 6 the variances of 119903 and

119887 are so small that we let 120590119903= 120590119887= 0 Then formula (A21)

becomes as follows

120588119881 = max119888

119906 (119888) + [119908 (1 minus 120579) (119903 + 119903) + 119908120579 (119887 +119887) minus 119888]119881

119908

+ (minus120577119903119903) 119881119903+ (minus120577

119887

119887)119881119887+

1

2

(120579120590119908)2

119881119908119908

(A22)

Differentiating formula (A22) with respect to 119903 and evaluat-ing at (119903 119887) = (0 0) we obtain 120588119881

119903= 119908(1 minus 120579)119881

119908+ (minus120577119903)119881119903+

(12)(120579120590119908)2

(120597119881119908119908120597119903) where 119881

119908119908= ((1 minus 120574)(minus120574)119908

2

)119881

which can be obtained from (22) then

120588119881119903= 119908 (1 minus 120579)119881

119908+ (minus120577119903) 119881119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119903

(A23)

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

Now differentiating (using the total derivative) formula(A23) with respect to 119908 and evaluating at (119903 119887) = (0 0) weobtain

120588119881119908119903= (1 minus 120579)119881

119908+ 119908 (1 minus 120579)119881

119908119908

+ (minus120577119903) 119881119908119903+

1

2

(1 minus 120574) (minus120574) 1205792

1205902

119881119908119903

(A24)

From formula (A23) we have

119881119903=

119908 (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A25)

According to formula (A24) and the term 119881119908119908

= (minus120574119908)119881119908

which can be obtained from (22) we have

119881119908119903=

(1 minus 120574) (1 minus 120579)119881119908

120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

21205902

(A26)

Let 120601(119888 120579 119908 119903 119887) denote the result of derivation of theright side in formula (A22) with respect to 119888 then 119888 satisfies120601 = 0 with 120601(119888 120579 119908 119903 119887) = 119906

119888minus 119881119908

Hence the impact of 119903 on 119888 is

119888119903= minus

120601119903

120601119888

(A27)

where

120601119888= 119906119888119888

120601119903= 119906119888119888

120597119888

120597119903

minus 119881119908119903

(A28)

Since all derivatives are evaluated at (119903 119887) = (0 0) we have119888 = 119888119889 120579 = 120579

119889 By substituting (A28) into (A27) and usingthe results of (22) (A25) (A26) now we can get

119888119903= minus

(1 minus 120574) (1 minus 120579119889

) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A29)

Similarly we have

119888119887= minus

(1 minus 120574) 119888119889

(120588 + 120577119887+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

(A30)

where the concrete expressions of 119888119889 120579119889 are referred toformula (22) 120579

119903 120579119887can be solved in an analogous way as

120579119903= minus1120574120590

2 120579119887= 1120574120590

2According to Proposition 7 119888119904 and 120579119904 can expressed as

119888119904

= 119888119889

+ 120591(minus

1 minus 120579119889

119872

120581120590119888

120590119903

) 119903 + 120591(minus

1

119873

120581120590119888

120590119887

)119887

120579119904

= 120579119889

+ 120591(minus

1

1205741205902

120581120590120579

120590119903

) 119903 + 120591(

1

1205741205902

120581120590120579

120590119887

)119887

(A31)

where

119872 =

(1 minus 120574) 119888119889

(120588 + 120577119903+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119873 =

(1 minus 120574) 119888119889

(120588 + 120577119906+ (12) 120574 (1 minus 120574) 120579

1198892

1205902) 120574

119888119889

= 119908(R (1 minus 120574))minus1120574

R =

[(120588 minus 119903 (1 minus 120574) minus (119887 minus 119903)

2

(1 minus 120574) 21205902

120574) 120574]

minus120574

(1 minus 120574)

120579119889

=

119887 minus 119903

1205741205902

(A32)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant nos 71371191 71210003 and712221061)

References

[1] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time caserdquo Review of Economics and Statisticsvol 51 no 3 pp 247ndash257 1969

[2] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[3] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science ampComplexity vol 22 no 3 pp 360ndash371 2009

[4] F Wen ldquoMeasuring and forecasting volatility in Chinese stockmarket usingHAR-CJ-MmodelrdquoAbstract andApplied Analysisvol 2013 Article ID 143194 13 pages 2013

[5] J C Cox andC-F Huang ldquoOptimal consumption and portfoliopolicies when asset prices follow a diffusion processrdquo Journal ofEconomic Theory vol 49 no 1 pp 33ndash83 1989

[6] I Karatzas J P Lehoczky S P Sethi and S E Shreve ldquoExplicitsolution of a general consumptioninvestment problemrdquoMath-ematics of Operations Research vol 11 no 2 pp 261ndash294 1986

[7] I Karatzas J P Lehoczky and S E Shreve ldquoOptimal portfolioand consumption decisions for a ldquosmall investorrdquo on a finitehorizonrdquo SIAM Journal on Control and Optimization vol 25no 6 pp 1557ndash1586 1987

[8] I Karatzas J P Lehoczky and S E Shreve ldquoExistence anduniqueness of multi-agent equilibrium in a stochastic dynamicconsumptioninvestment modelrdquo Mathematics of OperationsResearch vol 15 no 1 pp 80ndash128 1990

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

[9] R E Hall ldquoThe dynamic effects of fiscal policy in an economywith foresightrdquoThe Review of Economic Studies vol 38 no 114pp 229ndash244 1971

[10] M J Magill ldquoA local analysis of119873-sector capital accumulationunder uncertaintyrdquo Journal of EconomicTheory vol 15 no 1 pp211ndash219 1977

[11] F E Kydland and E C Prescott ldquoTime to build and aggregatefluctuationsrdquo Econometrica vol 50 no 6 pp 1345ndash1370 1982

[12] L J Christiano ldquoLinear-quadratic approximation and value-function iteration a comparisonrdquo Journal of Business amp Eco-nomic Statistics vol 8 no 1 pp 99ndash113 1990

[13] K L Judd and S-MGuu ldquoAsymptoticmethods for assetmarketequilibrium analysisrdquo Economic Theory vol 18 no 1 pp 127ndash157 2001

[14] G A Miller ldquoThe magical number seven plus or minustwo some limits on our capacity for processing informationrdquoPsychological Review vol 63 no 2 pp 81ndash97 1956

[15] D Kahneman Thinking Fast and Slow Macmillan New YorkNY USA 2011

[16] W Bi Y Sun H Liu and X Chen ldquoDynamic nonlinearpricing model based on adaptive and sophisticated learningrdquoMathematical Problems in Engineering vol 2014 Article ID791656 11 pages 2014

[17] P Bordalo N Gennaioli and A Shleifer ldquoSalience theory ofchoice under riskrdquoThe Quarterly Journal of Economics vol 127no 3 pp 1243ndash1285 2012

[18] B Koszegi and A Szeidl ldquoA model of focusing in economicchoicerdquo The Quarterly Journal of Economics vol 128 no 1 pp53ndash104 2013

[19] C A Sims ldquoImplications of rational inattentionrdquo Journal ofMonetary Economics vol 50 no 3 pp 665ndash690 2003

[20] L Peng and W Xiong ldquoInvestor attention overconfidence andcategory learningrdquo Journal of Financial Economics vol 80 no3 pp 563ndash602 2006

[21] M S Seasholes and G Wu ldquoPredictable behavior profits andattentionrdquo Journal of Empirical Finance vol 14 no 5 pp 590ndash610 2007

[22] BMackowiak andMWiederholt ldquoOptimal sticky prices underrational inattentionrdquo American Economic Review vol 99 no 3pp 769ndash803 2009

[23] X Gabaix ldquoA sparsity-based model of bounded rationalityapplied to basic consumer and equilibrium theoryrdquo WorkingPaper National Bureau of Economic Research YUNNewYorkNY USA 2013

[24] W H Fleming and H M Soner Controlled Markov Processesand Viscosity Solutions vol 25 Cambridge University PressCambridge UK 2006

[25] J-P Fouque G Papanicolaou and K R Sircar Derivatives inFinancial Markets with Stochastic Volatility Cambridge Univer-sity Press Cambridge UK 2000

[26] M I Kamien and N L Schwartz Dynamic Optimization TheCalculus of Variations and Optimal Control in Economics andManagement Courier Dover Publications Mineola NY USA1991

[27] F Wen Z He and X Chen ldquoInvestorsrsquo risk preference charac-teristics and conditional skewnessrdquo Mathematical Problems inEngineering vol 2014 Article ID 814965 14 pages 2014

[28] F Wen X Gong Y Chao and X Chen ldquoThe effects of prioroutcomes on risky choice evidence from the stock marketrdquoMathematical Problems in Engineering vol 2014 Article ID272518 8 pages 2014

[29] WH Fleming andT Pang ldquoAn application of stochastic controltheory to financial economicsrdquo SIAM Journal on Control andOptimization vol 43 no 2 pp 502ndash531 2004

[30] W H Fleming and D Hernandez-Hernandez ldquoAn optimalconsumption model with stochastic volatilityrdquo Finance andStochastics vol 7 no 2 pp 245ndash262 2003

[31] L Wang C X Dun W J Bi and Y R Zeng ldquoAn effectiveand efficient differential evolution algorithm for the integratedstochastic joint replenishment and delivery modelrdquoKnowledge-Based Systems vol 36 pp 104ndash114 2012

[32] L Wang C X Dun C G Lee Q L Fu and Y R ZengldquoModel and algorithm for fuzzy joint replenishment and deliv-ery scheduling without explicit membership functionrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 66 no 9ndash12 pp 1907ndash1920 2013

[33] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of