ThePrinciple-AgentConflictProbleminaContinuous-Time

Research ArticleThe Principle-Agent Conflict Problem in a Continuous-TimeDelegated Asset Management Model

Yanan Li and Chuanzheng Li

School of Finance Capital University of Economics and Business Beijing 100070 China

Correspondence should be addressed to Yanan Li 415758824qqcom

Received 9 July 2021 Revised 6 August 2021 Accepted 17 August 2021 Published 26 August 2021

Academic Editor Xiaofeng Zong

Copyright copy 2021 Yanan Li and Chuanzheng Li is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

is paper considers the principle-agent conflict problem in a continuous-time delegated asset management model when theinvestor and the fundmanager are all risk-averse with risk sensitivity coefficients cf and cm respectively Suppose that the investorentrusts his money to the fund manager e return of the investment is determined by the managerrsquos effort level and incentivestrategy but the benefit belongs to the investor In order to encourage the manager to work hard the investor will determine themanagerrsquos salary according to the terminal income is is a stochastic differential game problem and the distribution of incomebetween the manager and the investor is a key point to be solved in the custodymodele uncertain form of the incentive strategyimplies that the problem is different from the classical stochastic optimal control problem In this paper we first express theinvestorrsquos incentive strategy in term of two auxiliary processes and turn this problem into a classical one en we employ thedynamic programming principle to solve the problem

1 Introduction

Since professional asset management institutions can makeefficient investment decisions save investorsrsquo time and effortand simplify the investment process more andmore investorsnow entrust their money to fund managers securities firmsand other asset management organizations Nowadaysscholars pay more and more attention to asset managementproblems We can refer to [1ndash5] to name just a few

e whole asset management process involves twoparties the investor and the manager e return of theinvestment is closely related to the managerrsquos effort level andinvestment strategy but the interests belong to the investorSo the investor and managerrsquos relation poses a principal-agent conflict An important part of discussing the assetmanagement problem is finding the investorrsquos optimal in-centive mode under the principle agent conflict

ere are many papers committed to solving principal-agent conflict problems Most of the early literature studiesinvestigate the discrete-time case (we can refer to [6ndash8] or asummary book [9]) e problem in continuous-time

models is discussed for the first time in [10] It points outthat the investorrsquos optimal incentive mode is linear Seereferences [11ndash14] for further work In recent years themaximum principle or the martingale representation the-orem is often used to solve this problem in continuous-timemodels For the literature using the maximum principle wecan refer to [15 16] and for the literature of using themartingale representation theorem we can refer to [17 18]However since this problem often needs to solve a backwardstochastic differential equation (BSDE) that rarely has ex-plicit solutions there are few articles which give analyticalsolutions to this problem In order to get explicit solutions ofprincipal-agent conflict problems the authors of [19] ex-press the investorrsquos incentive strategy in terms of twoauxiliary processes and turn the principle agent probleminto a classical stochastic differential game problem

Although there are many papers committed to solvingprincipal-agent conflict problems in continuous-timemodels the delegated asset management problems areusually investigated in discrete-time models for the sake ofsimplicity us there are some contributions in this paper

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 3770868 10 pageshttpsdoiorg10115520213770868

(i) is paper considers the delegated asset manage-ment problem in a continuous-time model

(ii) Learning from [19] this paper gives explicit valuefunctions and the optimal strategies of both sides byexpressing the investorrsquos incentive strategy in termsof two auxiliary processes and turning the probleminto a classical stochastic differential game problem

(iii) In order to make the model more realistic thispaper brings in risk sensitivity coefficients to rep-resent the subjectsrsquo risk aversion attitudes

is paper is organized as follows In Section 2 weestablish a continuous-time model of the fund managementproblem In Section 3 we discuss the managerrsquos optimiza-tion problem under fixed investorrsquos incentive strategy Bysubstituting the managerrsquos optimal strategy into the inves-torrsquos optimal problem both the investor and the managerrsquosoptimal strategies are obtained in Section 4

2 The Principal-Agent Conflict Model

Similar to the model in [20] let us assume that the investoremploys a professional fund controller (manager) to investand the investor will get a profit and pay the manager at theterminal moment T Since the managerrsquos effort level cannotbe observed the investor will determine the managerrsquos salaryaccording to the terminal profit of the investment einvestorrsquos return is determined by the terminal investmentprofit and the managerrsquos salary e terminal investmentprofit is related to the managerrsquos investment strategy andeffort level and the incentive mechanism largely determinesthe managerrsquos strategy erefore the investor needs to findthe optimal incentive mechanism (the managerrsquos salary) tomaximize his terminal net income Meanwhile according tothe investorrsquos incentive mechanism the manager shall de-cide his investment strategy and the best effort level tomaximize his net salary (terminal salary minus effort cost)is is a non-cooperative game problem Next let us build amathematical model of this problem in probability space(ΩF P)

Similar to the model in [18] we suppose that themanagerrsquos effort will affect the fund income Rn

t whichsatisfies

dRnt R

nt r + μ + nt( 1113857dt + σdW(t)1113858 1113859 (1)

where μge 0 σ ge 0 and rgt 0 is the risk-free interest rateW(t) is a Brownian motion on (ΩF P) and nt1113864 1113865tge0 is themanagerrsquos effort level Here for the convenience of cal-culation we assume that the drift coefficient of Rn

t is a linearfunction of the managerrsquos effort level In fact as long as thedrift coefficient of Rn

t has the form of Rnt (r + f(nt)) for

some function f(n) the same method in this paper can beused after replacing n with f(n) For more general forms ofthe drift coefficient of Rn

t the existence of the time valuemakes it hard to obtain explicit solutions

Considering the managerrsquos strategy π (bπt nπt ) where

bπt represents the wealth that the manager decides to operateat moment t(e manager may not want to operate all the

wealth since the cost of the effort will increase with thewealth operated increases e money left will get a risk-freereturn) and nπ

t represents the managerrsquos effort level at t Bysome simple calculations we can get that the investmentincome under this strategy satisfies

dXπt rX

πt + b

πt μ + n

πt( 1113857( 1113857dt + b

πt σdW(t) (2)

Define the natural filtration produced by W(t) asFW

t1113864 1113865tge0 Now let us give the definition of both the managerand the investorrsquos admissible strategies Considering themanagerrsquos strategy π (bπt nπ

t ) If bπt and nπt are bounded

positive predictable stochastic processes under the strategyπ (2) has a unique solution

We call that strategy π (bπt nπt ) is admissible Denote

the set of all the managerrsquos admissible strategies by Π

Remark 1 Here we do not consider the case when b 0 orn 0 since in that case the model is meaningless

Suppose that the investorrsquos incentive strategy is afunction of the investment income at T and denote it byw(middot) If supπisinΠE[w(Xπ

T)]ltinfin the managerrsquos value functionunder w(middot) is a decreasing convex function with respect tothe initial wealth we say thatw(middot) is the investorrsquos admissiblestrategy Denote the set of all the investorrsquos admissiblestrategies by 1113954Π

Now let us analyze the whole game process Referring to[15] we know that investors play a leading role in the gameManagers need to decide their effort level and investmentstrategy according to the investorsrsquo incentive strategyerefore first we need to fix w(middot) and investigate themanagerrsquos optimal problem We can get the managerrsquosoptimal effort and investment strategy in terms of w(middot) as abyproduct en by substituting the managerrsquos optimalstrategy into the wealth process we can solve the investorrsquosoptimal problem by using the dynamic programmingprinciple

erefore firstly we fix the investorrsquos incentivestrategy w(middot) and consider the managerrsquos optimal problemSuppose that the manager is risk-averse and denote hisrisk sensitivity coefficient by cm lt 0 Referring to [18] wesuppose that the manager needs to pay (θn2b2) to manageb units of capital in unit time under the effort level n Hereθgt 0 is a constant which represents the effort cost pa-rameter e objective of the manager is to find the op-timal effort level and investment strategy to maximize hisnet income (salary minus effort cost) which is equivalentto minimize

Jπm(t x w) E e

cm w XπT( )minus 1113946

T

te

r(Tminus t) θ nπt( 1113857

221113872 1113873bπt dt1113888 1113889

|Xπt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)Denote the managerrsquos optimal strategy by πw then the

value function is

Vm(t x w) infπisinΠ

Jπm(t x w) J

πw

m (t x w) (4)

2 Mathematical Problems in Engineering

Suppose that the investor is risk-averse too his risk-sensitive coefficient is cf lt 0 Next we consider the inves-torrsquos optimal problem

If the managerrsquos salary is too high the investorrsquos incomewill be reduced If the managerrsquos salary is too low themanagerrsquos enthusiasm wanes which also deduces the in-vestorrsquos terminal income erefore the investor needs tofind a reasonable incentive strategy to maximize his netincome that is minimize

Jwf(t x) E e

cf XwT

minus w XwT( )( )|X

wt x1113876 1113877 (5)

where Xwt is the investment income process under strategy

πw us the investorrsquos value function is

Vf(t x) infwisin1113954Π

Jwf(t x) (6)

Remark 2 e problem discussed above is not a standardstochastic optimal control problem since the form of w(middot) isuncertain and we cannot solve it directly by using standardstochastic optimal methods In Section 3 we give anotherform of the incentive strategy and transform the gameproblem into a classical one en we can use the dynamicprogramming principle to solve the problem

3 The Managerrsquos Optimization Problem

Define Dt er(Tminus t) β(t π) cmDt(θnπ2t 2)bπt and

Γ(t T π) eminus 1113938

T

tβ(uπ)du en Jπm(t x w) can be denoted

by

Jπm(t x w) E Γ(t T π)e

cmw XπT( )|X

πt x1113876 1113877 (7)

Using the results of Section 34 in [21] we know thatunder the incentive strategy w(middot) the managerrsquos valuefunction Vm(t x w) satisfies the HJB equation

minus Vmt(t x w) infπisinΠ

minus β(t π)Vm(t x w) + rx + bπt μ + n

πt( 11138571113858 11138591113864

Vmx(t x w) +bπ2t σ2

2Vmxx(t x w)1113897

(8)

and the boundary condition

Vm(T x w) ecmw(x)

(9)

Since Vm(t x w) is a decreasing convex function of xfor forall(t x y z c) isin [0 T) times R times [0infin) times (minus infin 0) times (0infin)we can define the Hamiltonian function

H(t x y z c) infngt0bgt0

h(t x y z c n b) (10)

where

h(t x y z c n b) minus Dt

cmθn2b

2y +(rx + b(μ + n))z +

b2σ2

2c

(11)

Theorem 1

nlowastyzct

z

θcmyDt

(12)

blowastyzct

minus μ + nlowastyzct 2( 1113857( 1113857

σ2z

c (13)

is the minimum point of h in (10)

Proof According to the definition we know that h is aconvex function of (n b) So the minimum point of h in (10)is the stable point under constraint conditions ngt 0 bgt 0 Bysome simple calculations we have

hn(n b t x y z c) minus θDtbncmy + bz

hb(n b t x y z c) σ2cb +(μ + n)z minusDtθncmy

2

(14)

Combining the above two equations we can obtain thestable point of h

nlowastyzct

z

θcmyDt

gt 0

blowastyzct


σ2z

cgt 0

(15)

e proof is done

Remark 3 In this case the optimal investment strategy issimilar to that without principal-agent relationships eonly difference is that the numerator of the optimal in-vestment strategy is changed from (μ + n

lowastyzct ) into

(μ + (nlowastyzct 2)) Clearly this is due to the existence of the

agency relationship

Apparently the investorrsquos incentive strategy and themanagerrsquos value function are one-to-one In the followingwe will use auxiliary stochastic processes (Zt Γt) to deter-mine the managerrsquos value function and transform the in-vestorrsquos incentive strategy into (Zt Γt) en the problem inSection 2 can be translated into a classical stochastic optimalcontrol problem

First let us give the space of auxiliary stochastic pro-cesses (Z Γ) Fix t isin [0 T) let Z [t T] timesΩ⟶(minus infin 0) Γ [t T] timesΩ⟶ (0infin) be FW-predicable pro-cesses which satisfy

E 1113946T

tZ2sσ

2s + Γsσ

2s1113872 1113873ds1113890 1113891lt +infin (16)

Mathematical Problems in Engineering 3

Denote the set of all the processes satisfying the aboveconditions by V(t)

For some (Z Γ) isinV(t) and Yt ge 0 define theFW-progressively measurable process YZΓ on the filtrationspace (ΩF P FW

t1113864 1113865tge0) by

YZΓs Yt minus 1113946

s

tH r Xr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdXr +

12

1113946s

tΓrdlangXrangr s isin [t T]

(17)

where Xr is the investment income process Clearly forfixed Yt Z Γ YZΓ

T is only related to the investment incomeprocess and is FT measurable suppose that it is an in-centive strategy (we prove it in Corollary 1) In the fol-lowing we give the relationship between YZΓ

s and themanagerrsquos value function First we give the followinglemma

Lemma 1 Define

πlowastZΓ blowastZΓ

nlowastZΓ

1113872 1113873

blowastYZΓ

t ZtΓtt1113882 1113883

tge0 nlowastYZΓ

t ZtΓtt1113882 1113883

tge01113874 1113875

(18)

and then we have πlowastZΓ isin Π

Proof On the one hand since Z Γ YZΓ are all predictablestochastic processes referring to (12) and (13) we can getthat blowastZΓ and nlowastZΓ are bounded positive predictable sto-chastic processes On the other hand b

lowastyzct and n

lowastyzct are

independent of x Taking blowastZΓ and nlowastZΓ into (2) we can getthe Lipschitz continuity and linear growth of the coefficientsin (2) with respect to Xt then (2) has a unique solution eproof is done

Denote the investment income process under πlowastZΓ byXlowastZΓ We also have the following theorem

Theorem 2 Denote the managerrsquos value function with aterminal return (lnYZΓ

T cm) by Vm(t x YZΓT ) We can ob-

tain that

Yt Vm t x YZΓT1113872 1113873 (19)

Furthermore the managerrsquos optimal strategy is πlowastZΓ

Proof forallπ isin Π s isin [t T] we have


s

tH r X

πr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdX

πr +

12

1113946s

tΓrdlangX

πrangr

(20)

Using Itorsquos formula we have

deminus 1113946

r

tβ(u π)du

YZΓr e

minus 1113946r

tβ(u π)du

minus H r Xπr Y

ZΓr Zr Γr1113872 11138731113960

+ rXπr + b

πr μ + n

πr( 1113857( 1113857Zr

+bπ2r σ2

2Γr minus β(r π)1113891dr

+ eminus 1113946

r

tβ(u π)du

σZrdW(r)

(21)

It follows from (16) that eminus 1113938

r

tβ(uπ)duσZrdW(r) is a

martingale Integrating and taking expectations on bothsides of (21) we can get

Yt geE eminus 1113946

T

tβ(u π)du

YZΓT |X

πt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ J

πm t x Y

ZΓT1113872 1113873

(22)

Furthermore by simple calculations under πlowastZΓ isin Πwe have

dYZΓt β t πlowastZΓ

1113872 1113873YZΓt dt + b

lowastYZΓt ZtΓt

t ZtσdWt (23)

Using (23) and Itorsquos formula we can obtain

deminus 1113946

r

tβ u πlowastZΓ

1113872 1113873duY

ZΓr e

minus 1113946r

tβ u πlowastZΓ

1113872 1113873dublowastYZΓ

t ZtΓtt ZtσdWt

(24)

With similar methods integrating and taking expecta-tions on both sides of (24) we have

Yt E eminus 1113946

T

tβ u πlowastZΓ

1113872 1113873duY

ZΓT |XlowastZΓt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

JπlowastZΓ

m t x YZΓT1113872 1113873ge J

πm t x Y

ZΓT1113872 1113873

(25)

is implies that πlowastZΓ is the managerrsquos optimal strategyand

Yt Vm t x YZΓT1113872 1113873 (26)

Up till now fixing (Z Γ) isin V(t) we can get themanagerrsquos optimal strategy and represent the managerrsquosvalue function In Section 4 we begin to consider the in-vestorrsquos optimization problem at is finding the optimal(Z Γ) isin V(t) to maximize the investorrsquos net profit

4 The Investorrsquos Optimization Problem

Suppose that the investorrsquos wealth is x at t Apparently theinvestorrsquos value function is uniquely determined by thewealth process and the managerrsquos value function So the


objective of the investor is to find the optimal (Z Γ) isinV(t)

to minimize his value function Define

v(t x y) inf(ZΓ)isinV(t)

E ecf XlowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

πt x Y

ZΓt y1113876 1113877

(27)

Referring to eorem 41 in [19] we know that if As-sumption 32 Assumption 43 and Assumption 44 in [19]hold the investorrsquos value function satisfies

Vf(t x) infyisin 0ecmR[ ]

v(t x y) (28)

Here R is the minimum pay in order to make sure thatthe manager takes the job

Section 41 gives the verification of the threeassumptions

41 7e Verification of Assumptions

Assumption 1 (Assumption 32 in [19]) H has at least oneextreme point (b

lowastyzct n

lowastyzct ) For any t isin [0 T]

(Z Γ) isin V(t) we have πlowastZΓ isin Π

Proof is is the result of eorem 1 and Lemma 1e Hamiltonian function can be expressed as

H(t x y z c) infbgt0

F(t x y z b) +b2σ2

2c1113896 1113897 (29)

Here

F(t x y z b) infngt0

minus Dt

cmθn2b

2y +(rx + b(μ + n))z1113896 1113897

(30)

Define

YZs Yt minus 1113946

s

tF r Xr Y

Zr Zr1113872 1113873dr + 1113946

s

tZrdXr s isin [t T]

(31)

and we have the following assumption

Assumption 2 (Assumption 43 in [19]) F has at least oneextreme point n

lowastyzbt furthermore (b nlowastY

ZZb) isin Π

Proof On the one hand the right hand of F is a parabolawith an opening up with respect to n so the minimum pointis attained at the axis of the parabola (zDtcmθy) that isnlowastyzbt (zDtcmθy) On the other hand since Zlt 0 is

predictable we can get that nlowastYZ

t Ztbt (bZtDtcmθbYZ

t ) is apositive predictable process Furthermore b and n

lowastyzbt are

independent of x is implies the Lipschitz continuity andlinear growth of the coefficients in (2) with respect to theinvestment income process then (2) has a uniquesolution

Assumption 3 (Assumption 44 in [19]) forallbgt 0 (1b2σ2) isbounded

Proof We can get the result directly from σ gt 0 bgt 0

427e Investorrsquos Value Function Clearly as soon as we getv(t x y) we can obtain Vf(t x) e following theoremgives the partial differential equation satisfied by v(t x y)

Theorem 3 v(t x y) is the viscosity solution of

minus vt(t x y) inf(ZΓ)isinV(t)

G(t x y Z Γ) (32)

v(T x y) ecfx

yminus cfcm( 1113857

(33)

where

G(t x y Z Γ) rx + blowastZΓt μ + n

lowastZΓt1113872 11138731113960 1113961vx

+σ2 blowastZΓt1113872 1113873

2

2vxx +

Dtcmθ nlowastZΓt1113872 1113873

2

2blowastZΓt yvy

+σ2 blowastZΓt1113872 1113873

2

2Z2vyy + σ2 b

lowastZΓt1113872 1113873

2Zvxy

(34)

Proof By the definition of v(t x y) we can obtain that itsatisfies (33) Furthermore according to the dynamic pro-gramming principle we have


v t + h XlowastZΓt+h Y

ZΓt+h1113872 1113873 (35)

By using Itorsquos formula with respect to v(s XlowastZΓs YZΓ

s )

from t to t + h we have


ZΓt+h1113872 1113873 v(t x y) + 1113946

t+h

tvt s X

lowastZΓs Y

ZΓs1113872 1113873

+ G s XlowastZΓs Y

ZΓs Zs Γs1113872 1113873ds

(36)

Combining with the above two equations we can get

vt(t x y) + inf(ZΓ)isinV(t)

G(t x y Z Γ) 0 (37)

at is v(t x y) satisfies (32) e proof is doneNext we are going to solve (32) and (33) Considering

the boundary condition we guess

v(t x y) ecfDtxy

minus cfcm( 1113857E(t) (38)

where E(t) is a function of t which satisfies E(T) 1If the variables in the solution can be separated from

each other (32) can be easily solved However (32) containsecfDtx which is a cross term of t and x To cancel the crossterm we introduce zt DtX

lowastZΓt Using Itorsquos formula we

can get

dzt minus rDtXlowastZΓt dt + DtdX

lowastZΓt

DtblowastZΓt μ + n

lowastZΓt1113872 1113873dt + σdW(t)1113960 1113961

(39)


We can also obtain zT XlowastZΓT Define

V(t z y) inf(ZΓ)isinV(t)

E ecf zTminus lnYZΓ

Tcm( )( )|zt z1113876 1113877

inf(ZΓ)isinV(t)

E ecf X lowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

lowastZΓt

z

Dt

1113890 1113891

v tz

Dt

y1113888 1113889

(40)

Obviously solving v(t x y) is equivalent to solvingV(t z y) Using a similar method as the one in eorem 3we can get that

minus Vt inf(ZΓ)isinV(t)

minusμ + n

lowastZΓt 21113872 11138731113872 1113873 μ + n

lowastZΓt1113872 1113873

σ2Dt

Z

ΓVz

⎧⎨

⎩

minuscmθ n

lowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2σ2yDt

Z

ΓVy

+μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2D

2t

Z2

Γ2Vzz

+nlowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2cmθyDt( 1113857

2Z2

Γ2Vyy

+nlowastZΓt μ + n

lowastZΓt 21113872 11138731113872 1113873

2

σ2cmθyD

2t

Z2

Γ2Vzy

⎫⎪⎬

⎪⎭

(41)

V(T z y) ecfz


(42)

e first step in solving (41) is to find its minimum pointDefine MZΓ (ZΓ) it is shown in Section 3 that (Z Γ) and(MZΓ nlowastZΓ) are one-to-one en (41) is transformed into

minus Vt inf(nM)isinR+timesRminus

minus(μ +(n2))(μ + n)

σ2DtMVz1113896

+(μ +(n2))

2

2σ2D

2t M

2Vzz minus

cmθn2(μ +(n2))

2σ2yDtMVy

+n2(μ +(n2))

2

2σ2cmθyDt( 1113857

2M

2Vyy

+n(μ +(n2))

2

σ2cmθyD

2t M

2Vzy1113897

(43)

Now the problem of finding the minimum point in (41) ischanged into a problem of finding theminimumpoint in (43)

According to (38) we suppose thatV(t z y) E(t)ecfzyminus (cfcm) By some simple calculationswe can get that

Vz(t z y) cfV(t z y)

Vzz(t z y) c2fV(t z y)

yVy(t z y) minuscf

cm

V(t z y)

y2Vyy(t z y)

cf cf + cm1113872 1113873

c2m

V(t z y)

yVzy(t z y) minusc2f

cm

V(t z y)

(44)

Taking them into (43) we have

minus Eprime(t)V(t z y) inf(nM)isinR+timesRminus


σ2DtMcf1113896

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

21113897E(t)V(t z y)

(45)

Since the right hand of (45) is continuous the minimumpoint can only be attained at the stable points or theboundary points which depends on the parameter valuesDenote the minimum point of (45) by (nlowastt Mlowastt ) and denotethe corresponding minimum point of (41) by (Zlowastt Γlowastt ) It isshown from the Appendix that nlowastt and DtM

lowastt are constants

concerning μ θ cf and cm Let nlowastt nlowast

Remark 4 On the one hand the exponential form of theobjective function implies that blowastt is independent of Xlowastt Onthe other hand the benefit and the cost brought by themanagerrsquos effort are only related to blowastt so nlowastt is independentof Xlowastt Furthermore in this paper we consider the dis-counted benefit and cost brought by the managerrsquos effort sonlowastt is independent of t

Remark 5 It is shown from figures in the Appendix that nlowast

decreases with an increase in μ(the drift coefficient of thefund wealth process) θ(the effort cost coefficient) and|cm|(the managerrsquos risk aversion level) It increases with anincrease in |cf|(the investorrsquos risk aversion level)


Remark 6 Define Ylowastt Vm(t x YZlowastΓlowastT ) considering (12)

and (13) we can get that (Zlowastt Ylowastt ) θcmDtnlowast and Dtb

lowastt

(minus (μ + (nlowast2))σ2)DtMlowastt are constants

Taking the minimum point into (45) and solving it wecan get

V(t z y) eB(Tminus t)

ecfz


(46)

Here

B minusμ + n

lowast2( 1113857( 1113857 μ + nlowast

( 1113857

σ2DtMlowastt cf

+μ + n

lowast2( 1113857( 11138572

2σ2D

2t Mlowast 2t c

2f +

cfθnlowast2 μ + n

lowast2( 1113857( 1113857

2σ2DtMlowastt

+nlowast2 μ + n

lowast2( 1113857( 11138572

2σ2θ2D2

t Mlowast 2t cf cf + cm1113872 1113873

minusnlowast μ + n

lowast2( 1113857( 11138572

σ2c2fθD

2t Mlowast 2t

(47)

0 000

002

004

006

008

010

n

035 040 045030the drift coefficient of the fund

0ndash

000

002

004

006

008

010

n

20 25 3010 15The effort cost coefficient

0

000

002

004

006

008

010

n

020 025 030015The managerrsquos risk aversion level

0

011 012 013 014 015010The investorrsquos risk aversion level

000

002

004

006

008

010

n

Figure 1 e effects of parameters on the optimal effort level


is a constant As a consequence we can also get the followingresults

v(t x y) eB(Tminus t)

ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)

43 7e Investorrsquos Excitation Mechanism In this section letus analyze the investorrsquos excitation mechanism DenoteYlowastt YZlowastΓlowast

t From the above analysis we know that

dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b

lowastt Zlowastt σdWt

Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt

σdWt minus blowast 2t σ2

Zlowast 2t

2Ylowast 2t

dt

(50)

Furthermore we can get that the investment incomeunder nlowast and blowastt satisfies

dXlowastt rX

lowastt + blowastt nlowast

+ μ( 1113857( 1113857dt + blowastt σdWt (51)

which implies

d lnYlowastt

Dtcmθnlowast2

blowastt

2minus blowast 2t σ2

Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm

Dtblowastt θnlowast2

2minus12D

2t blowast 2t θ2cmn

lowast2σ2 + Dtblowastt θnlowast

nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant

A Dtblowastt θnlowast2

2minus12D



nlowast

+ μ( 1113857gt 0

(53)

and then we can obtaind lnY

lowastt cmAdt + cmθn

lowastDt dX

lowastt minus rX

lowastt dt( 1113857 cm Adt + θn

lowastdDtXlowastt( 1113857

(54)

So

lnYlowastT minus cmR cm A(T minus t) + n

lowastθ XlowastT minus Dtx( 11138571113858 1113859 (55)

can be deduced immediately Since lnYT cmw(XT) wecan get the strategy

w XT( 1113857 A(T minus t) + nlowastθ XlowastT minus Dtx( 1113857 + R (56)

It is a linear function of XlowastT minus Dtx which is the dis-counted profit of the investment

Remark 7 It follows from the above results that the man-agerrsquos wages increase with the increase of the cost coefficientthe effort level and the discounted profit of the investmentFurthermore the longer the work the higher the salary It isconsistent with reality

We can also get the following corollary

Corollary 1

Ylowasts Vm s X

lowasts YlowastT( 1113857 e

cm R+A(Tminus t)+nlowastθ Xlowasts minus Dtx( )[ ] (57)

is implies that Vm(s Xlowasts YlowastT) is a decreasing convexfunction of Xlowasts us the assumption in Section 2 that YlowastT isan incentive strategy is proved

Appendix

Define

I(n M t) minus(μ +(n2))(μ + n)

σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)

We know that there are three kinds of points which maybe the minimum point of (45)


(i) e points which satisfy In(n M t) 0 IM

(n M t) 0(ii) e points which satisfy n 0 IM(0 M t) 0(iii) e points which satisfy M 0 In(n 0 t) 0

With parameters fixed we can easily decide which is theminimum point of (45) In the following we will investigatethe form of those points

e first kind of points (n1t M1t) is the solution of thefollowing equations

In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)

We can deduce from (A2) that

DtM1t n1t + μ minus n

21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf minus cf21113872 1113873 + cfθμ minus cfθn1t21113872 1113873 + θ2μ cf + cm1113872 1113873n1t1113960 1113961

(A4)

It also follows from (A3) that

DtM1t n1t + μ minus n1t( 1113857

2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)

Combining the above two equations we can get that

n1t + μ minusn1t( 1113857

2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)

Clearly by solving (A5) and (A6) we can get that n1 andDtM1t are constants

Denote the second kind of point by (0 M2t) us M2t

satisfies (A5) with n replaced with 0 and we can get thatDtM2t is a constant

Denote the third kind of points by (n3t 0) ey satisfy(A4) By solving it we can get that n3t is a constant

Denote the minimum point of (45) by (nlowastt Mlowastt ) Itfollows from the above analysis that nlowastt and DtM

lowastt are all

constants For different μ θ cf and cm by calculating (A6)(A2) or (A3) we can get different nlowast

By using R we plot the following figures which indicatethe effect of μ θ cf and cm on nlowast (Figure 1)

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (11901404)

References

[1] A Almazan K C Brown M Carlson and D A ChapmanldquoWhy constrain your mutual fund managerrdquo Journal of Fi-nancial Economics vol 73 no 2 pp 289ndash321 2004

[2] M K Brunnermeier and L H Pedersen ldquoMarket liquidityand funding liquidityrdquo Review of Financial Studies vol 22no 6 pp 2201ndash2238 2009

[3] P H Dybvig H K Farnsworth and J N CarpenterldquoPortfolio performance and agencyrdquo Review of FinancialStudies vol 23 no 1 pp 1ndash23 2010


[4] S Gervais A W Lynch and D K Musto ldquoFund families asdelegated monitors of money managersrdquo Review of FinancialStudies vol 18 no 4 pp 1139ndash1169 2005

[5] V Guerrieri and P Kondor ldquoFund managers career con-cerns and asset price volatilityrdquo 7e American EconomicReview vol 102 no 5 pp 1986ndash2017 2012

[6] S Ross ldquoe economic theory of agency the principalrsquosproblemrdquo 7e American Economic Review vol 63 pp 134ndash139 1973

[7] J A Mirrlees ldquoe optimal structure of incentives and au-thority within an organizationrdquo7eBell Journal of Economicsvol 7 no 1 pp 105ndash131 1976

[8] B Holmstrom ldquoMoral hazard and observabilityrdquo 7e BellJournal of Economics vol 10 no 1 pp 74ndash91 1979

[9] P Bolton and M Dewatripont Contract 7eory MIT PressCambridge UK 2005

[10] B Holmstrom and P Milgrom ldquoAggregation and linearity inthe provision of intertemporal incentivesrdquo Econometricavol 55 no 2 pp 303ndash328 1987

[11] H Schattler and J Sung ldquoe first-order approach to thecontinuous-time principal-agent problem with exponentialutilityrdquo Journal of Economic7eory vol 61 no 2 pp 331ndash3711993

[12] H Schattler and J Sung ldquoOn optimal sharing rules in dis-crete-and continuous-time principal-agent problems withexponential utilityrdquo Journal of Economic Dynamics andControl vol 21 no 2-3 pp 551ndash574 1997

[13] H M Muller ldquoe first-best sharing rule in the continuous-time principal-agent problem with exponential utilityrdquoJournal of Economic 7eory vol 79 no 2 pp 276ndash280 1998

[14] H M Muller ldquoAsymptotic efficiency in dynamic principal-agent problemsrdquo Journal of Economic 7eory vol 91 no 2pp 292ndash301 2000

[15] J Cvitanic X Wan and J Zhang ldquoOptimal compensationwith hidden action and lump-sum payment in a continuous-time modelrdquo Applied Mathematics and Optimization vol 59pp 99ndash146 2009

[16] J Cvitanic and J Zhang Contract 7eory in Continuous TimeModels Springer-Verlag Berlin Germany 2013

[17] Y Sannikov ldquoA continuous-time version of the principal-agent problemrdquo7eReview of Economic Studies vol 75 no 3pp 957ndash984 2008

[18] Z He ldquoA Model of dynamic compensation and capitalstructurerdquo Journal of Financial Economics vol 100 no 2pp 351ndash366 2011

[19] J Cvitanic D Possamai and N Touzi ldquoDynamic pro-gramming approach to principal-agent problemsrdquo Financeand Stochastics vol 22 pp 1ndash37 2018

[20] Z He and W Xiong ldquoDelegated asset management invest-ment mandates and capital immobilityrdquo Journal of FinancialEconomics vol 107 no 2 pp 239ndash258 2013

[21] H Pham Continuous-time Stochastic Control and Optimi-zation with Financial Applications Springer-Verlag NewYork NY USA 2009


(i) is paper considers the delegated asset manage-ment problem in a continuous-time model

(ii) Learning from [19] this paper gives explicit valuefunctions and the optimal strategies of both sides byexpressing the investorrsquos incentive strategy in termsof two auxiliary processes and turning the probleminto a classical stochastic differential game problem

(iii) In order to make the model more realistic thispaper brings in risk sensitivity coefficients to rep-resent the subjectsrsquo risk aversion attitudes

is paper is organized as follows In Section 2 weestablish a continuous-time model of the fund managementproblem In Section 3 we discuss the managerrsquos optimiza-tion problem under fixed investorrsquos incentive strategy Bysubstituting the managerrsquos optimal strategy into the inves-torrsquos optimal problem both the investor and the managerrsquosoptimal strategies are obtained in Section 4

2 The Principal-Agent Conflict Model

Similar to the model in [20] let us assume that the investoremploys a professional fund controller (manager) to investand the investor will get a profit and pay the manager at theterminal moment T Since the managerrsquos effort level cannotbe observed the investor will determine the managerrsquos salaryaccording to the terminal profit of the investment einvestorrsquos return is determined by the terminal investmentprofit and the managerrsquos salary e terminal investmentprofit is related to the managerrsquos investment strategy andeffort level and the incentive mechanism largely determinesthe managerrsquos strategy erefore the investor needs to findthe optimal incentive mechanism (the managerrsquos salary) tomaximize his terminal net income Meanwhile according tothe investorrsquos incentive mechanism the manager shall de-cide his investment strategy and the best effort level tomaximize his net salary (terminal salary minus effort cost)is is a non-cooperative game problem Next let us build amathematical model of this problem in probability space(ΩF P)

Similar to the model in [18] we suppose that themanagerrsquos effort will affect the fund income Rn

t whichsatisfies

dRnt R

nt r + μ + nt( 1113857dt + σdW(t)1113858 1113859 (1)

where μge 0 σ ge 0 and rgt 0 is the risk-free interest rateW(t) is a Brownian motion on (ΩF P) and nt1113864 1113865tge0 is themanagerrsquos effort level Here for the convenience of cal-culation we assume that the drift coefficient of Rn

t is a linearfunction of the managerrsquos effort level In fact as long as thedrift coefficient of Rn

t has the form of Rnt (r + f(nt)) for

some function f(n) the same method in this paper can beused after replacing n with f(n) For more general forms ofthe drift coefficient of Rn

t the existence of the time valuemakes it hard to obtain explicit solutions

Considering the managerrsquos strategy π (bπt nπt ) where

bπt represents the wealth that the manager decides to operateat moment t(e manager may not want to operate all the

wealth since the cost of the effort will increase with thewealth operated increases e money left will get a risk-freereturn) and nπ

t represents the managerrsquos effort level at t Bysome simple calculations we can get that the investmentincome under this strategy satisfies

dXπt rX

πt + b

πt μ + n

πt( 1113857( 1113857dt + b

πt σdW(t) (2)

Define the natural filtration produced by W(t) asFW

t1113864 1113865tge0 Now let us give the definition of both the managerand the investorrsquos admissible strategies Considering themanagerrsquos strategy π (bπt nπ

t ) If bπt and nπt are bounded

positive predictable stochastic processes under the strategyπ (2) has a unique solution

We call that strategy π (bπt nπt ) is admissible Denote

the set of all the managerrsquos admissible strategies by Π

Remark 1 Here we do not consider the case when b 0 orn 0 since in that case the model is meaningless

Suppose that the investorrsquos incentive strategy is afunction of the investment income at T and denote it byw(middot) If supπisinΠE[w(Xπ

T)]ltinfin the managerrsquos value functionunder w(middot) is a decreasing convex function with respect tothe initial wealth we say thatw(middot) is the investorrsquos admissiblestrategy Denote the set of all the investorrsquos admissiblestrategies by 1113954Π

Now let us analyze the whole game process Referring to[15] we know that investors play a leading role in the gameManagers need to decide their effort level and investmentstrategy according to the investorsrsquo incentive strategyerefore first we need to fix w(middot) and investigate themanagerrsquos optimal problem We can get the managerrsquosoptimal effort and investment strategy in terms of w(middot) as abyproduct en by substituting the managerrsquos optimalstrategy into the wealth process we can solve the investorrsquosoptimal problem by using the dynamic programmingprinciple

erefore firstly we fix the investorrsquos incentivestrategy w(middot) and consider the managerrsquos optimal problemSuppose that the manager is risk-averse and denote hisrisk sensitivity coefficient by cm lt 0 Referring to [18] wesuppose that the manager needs to pay (θn2b2) to manageb units of capital in unit time under the effort level n Hereθgt 0 is a constant which represents the effort cost pa-rameter e objective of the manager is to find the op-timal effort level and investment strategy to maximize hisnet income (salary minus effort cost) which is equivalentto minimize

Jπm(t x w) E e

cm w XπT( )minus 1113946

T

te

r(Tminus t) θ nπt( 1113857

221113872 1113873bπt dt1113888 1113889

|Xπt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)Denote the managerrsquos optimal strategy by πw then the

value function is

Vm(t x w) infπisinΠ

Jπm(t x w) J

πw

m (t x w) (4)




Jwf(t x) E e

cf XwT

minus w XwT( )( )|X

wt x1113876 1113877 (5)




Jwf(t x) (6)





T


by


cmw XπT( )|X

πt x1113876 1113877 (7)




πt( 11138571113858 11138591113864


2Vmxx(t x w)1113897

(8)


Vm(T x w) ecmw(x)

(9)



h(t x y z c n b) (10)

where


cmθn2b

2y +(rx + b(μ + n))z +

b2σ2

2c

(11)

Theorem 1

nlowastyzct

z

θcmyDt

(12)

blowastyzct


σ2z

c (13)





2

(14)


nlowastyzct

z

θcmyDt

gt 0

blowastyzct


σ2z

cgt 0

(15)

e proof is done


lowastyzct ) into


agency relationship



E 1113946T

tZ2sσ

2s + Γsσ

2s1113872 1113873ds1113890 1113891lt +infin (16)




t1113864 1113865tge0) by


s

tH r Xr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdXr +

12

1113946s


(17)




Lemma 1 Define


nlowastZΓ

1113872 1113873

blowastYZΓ

t ZtΓtt1113882 1113883

tge0 nlowastYZΓ

t ZtΓtt1113882 1113883

tge01113874 1113875

(18)



lowastyzct and n

lowastyzct are





tain that

Yt Vm t x YZΓT1113872 1113873 (19)




s

tH r X

πr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdX

πr +

12

1113946s

tΓrdlangX

πrangr

(20)


deminus 1113946

r

tβ(u π)du

YZΓr e

minus 1113946r

tβ(u π)du

minus H r Xπr Y

ZΓr Zr Γr1113872 11138731113960

+ rXπr + b

πr μ + n

πr( 1113857( 1113857Zr

+bπ2r σ2


+ eminus 1113946

r

tβ(u π)du

σZrdW(r)

(21)


r




T

tβ(u π)du

YZΓT |X

πt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ J

πm t x Y

ZΓT1113872 1113873

(22)



1113872 1113873YZΓt dt + b

lowastYZΓt ZtΓt

t ZtσdWt (23)


deminus 1113946

r

tβ u πlowastZΓ

1113872 1113873duY

ZΓr e

minus 1113946r

tβ u πlowastZΓ


t ZtΓtt ZtσdWt

(24)


Yt E eminus 1113946

T

tβ u πlowastZΓ

1113872 1113873duY

ZΓT |XlowastZΓt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

JπlowastZΓ

m t x YZΓT1113872 1113873ge J

πm t x Y

ZΓT1113872 1113873

(25)


Yt Vm t x YZΓT1113872 1113873 (26)








E ecf XlowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

πt x Y

ZΓt y1113876 1113877

(27)



v(t x y) (28)





lowastyzct n





F(t x y z b) +b2σ2

2c1113896 1113897 (29)

Here


minus Dt

cmθn2b

2y +(rx + b(μ + n))z1113896 1113897

(30)

Define


s

tF r Xr Y

Zr Zr1113872 1113873dr + 1113946

s

tZrdXr s isin [t T]

(31)




ZZb) isin Π





lowastyzbt are







G(t x y Z Γ) (32)

v(T x y) ecfx


(33)

where


lowastZΓt1113872 11138731113960 1113961vx

+σ2 blowastZΓt1113872 1113873

2

2vxx +


2

2blowastZΓt yvy

+σ2 blowastZΓt1113872 1113873

2

2Z2vyy + σ2 b

lowastZΓt1113872 1113873

2Zvxy

(34)




ZΓt+h1113872 1113873 (35)


s )



ZΓt+h1113872 1113873 v(t x y) + 1113946

t+h

tvt s X

lowastZΓs Y

ZΓs1113872 1113873

+ G s XlowastZΓs Y

ZΓs Zs Γs1113872 1113873ds

(36)



G(t x y Z Γ) 0 (37)



v(t x y) ecfDtxy

minus cfcm( 1113857E(t) (38)




can get


lowastZΓt



(39)





Tcm( )( )|zt z1113876 1113877

inf(ZΓ)isinV(t)

E ecf X lowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

lowastZΓt

z

Dt

1113890 1113891

v tz

Dt

y1113888 1113889

(40)



minusμ + n

lowastZΓt 21113872 11138731113872 1113873 μ + n

lowastZΓt1113872 1113873

σ2Dt

Z

ΓVz

⎧⎨

⎩

minuscmθ n

lowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2σ2yDt

Z

ΓVy

+μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2D

2t

Z2

Γ2Vzz

+nlowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2cmθyDt( 1113857

2Z2

Γ2Vyy

+nlowastZΓt μ + n

lowastZΓt 21113872 11138731113872 1113873

2

σ2cmθyD

2t

Z2

Γ2Vzy

⎫⎪⎬

⎪⎭

(41)

V(T z y) ecfz


(42)




σ2DtMVz1113896

+(μ +(n2))

2

2σ2D

2t M

2Vzz minus

cmθn2(μ +(n2))

2σ2yDtMVy

+n2(μ +(n2))

2

2σ2cmθyDt( 1113857

2M

2Vyy

+n(μ +(n2))

2

σ2cmθyD

2t M

2Vzy1113897

(43)





yVy(t z y) minuscf

cm

V(t z y)

y2Vyy(t z y)

cf cf + cm1113872 1113873

c2m

V(t z y)


cm

V(t z y)

(44)




σ2DtMcf1113896

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

21113897E(t)V(t z y)

(45)










lowastt




ecfz


(46)

Here

B minusμ + n

lowast2( 1113857( 1113857 μ + nlowast

( 1113857

σ2DtMlowastt cf

+μ + n

lowast2( 1113857( 11138572

2σ2D

2t Mlowast 2t c

2f +

cfθnlowast2 μ + n

lowast2( 1113857( 1113857

2σ2DtMlowastt

+nlowast2 μ + n

lowast2( 1113857( 11138572

2σ2θ2D2


minusnlowast μ + n

lowast2( 1113857( 11138572

σ2c2fθD

2t Mlowast 2t

(47)

0 000

002

004

006

008

010

n


0ndash

000

002

004

006

008

010

n


0

000

002

004

006

008

010

n


0


000

002

004

006

008

010

n





ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)



dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b


Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt


Zlowast 2t

2Ylowast 2t

dt

(50)


dXlowastt rX



which implies

d lnYlowastt

Dtcmθnlowast2

blowastt


Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm


2minus12D



nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant


2minus12D



nlowast

+ μ( 1113857gt 0

(53)



lowastDt dX

lowastt minus rX



(54)

So








Corollary 1

Ylowasts Vm s X




Appendix

Define


σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)







In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References


























Jwf(t x) E e

cf XwT

minus w XwT( )( )|X

wt x1113876 1113877 (5)




Jwf(t x) (6)





T


by


cmw XπT( )|X

πt x1113876 1113877 (7)




πt( 11138571113858 11138591113864


2Vmxx(t x w)1113897

(8)


Vm(T x w) ecmw(x)

(9)



h(t x y z c n b) (10)

where


cmθn2b

2y +(rx + b(μ + n))z +

b2σ2

2c

(11)

Theorem 1

nlowastyzct

z

θcmyDt

(12)

blowastyzct


σ2z

c (13)





2

(14)


nlowastyzct

z

θcmyDt

gt 0

blowastyzct


σ2z

cgt 0

(15)

e proof is done


lowastyzct ) into


agency relationship



E 1113946T

tZ2sσ

2s + Γsσ

2s1113872 1113873ds1113890 1113891lt +infin (16)




t1113864 1113865tge0) by


s

tH r Xr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdXr +

12

1113946s


(17)




Lemma 1 Define


nlowastZΓ

1113872 1113873

blowastYZΓ

t ZtΓtt1113882 1113883

tge0 nlowastYZΓ

t ZtΓtt1113882 1113883

tge01113874 1113875

(18)



lowastyzct and n

lowastyzct are





tain that

Yt Vm t x YZΓT1113872 1113873 (19)




s

tH r X

πr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdX

πr +

12

1113946s

tΓrdlangX

πrangr

(20)


deminus 1113946

r

tβ(u π)du

YZΓr e

minus 1113946r

tβ(u π)du

minus H r Xπr Y

ZΓr Zr Γr1113872 11138731113960

+ rXπr + b

πr μ + n

πr( 1113857( 1113857Zr

+bπ2r σ2


+ eminus 1113946

r

tβ(u π)du

σZrdW(r)

(21)


r




T

tβ(u π)du

YZΓT |X

πt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ J

πm t x Y

ZΓT1113872 1113873

(22)



1113872 1113873YZΓt dt + b

lowastYZΓt ZtΓt

t ZtσdWt (23)


deminus 1113946

r

tβ u πlowastZΓ

1113872 1113873duY

ZΓr e

minus 1113946r

tβ u πlowastZΓ


t ZtΓtt ZtσdWt

(24)


Yt E eminus 1113946

T

tβ u πlowastZΓ

1113872 1113873duY

ZΓT |XlowastZΓt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

JπlowastZΓ

m t x YZΓT1113872 1113873ge J

πm t x Y

ZΓT1113872 1113873

(25)


Yt Vm t x YZΓT1113872 1113873 (26)








E ecf XlowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

πt x Y

ZΓt y1113876 1113877

(27)



v(t x y) (28)





lowastyzct n





F(t x y z b) +b2σ2

2c1113896 1113897 (29)

Here


minus Dt

cmθn2b

2y +(rx + b(μ + n))z1113896 1113897

(30)

Define


s

tF r Xr Y

Zr Zr1113872 1113873dr + 1113946

s

tZrdXr s isin [t T]

(31)




ZZb) isin Π





lowastyzbt are







G(t x y Z Γ) (32)

v(T x y) ecfx


(33)

where


lowastZΓt1113872 11138731113960 1113961vx

+σ2 blowastZΓt1113872 1113873

2

2vxx +


2

2blowastZΓt yvy

+σ2 blowastZΓt1113872 1113873

2

2Z2vyy + σ2 b

lowastZΓt1113872 1113873

2Zvxy

(34)




ZΓt+h1113872 1113873 (35)


s )



ZΓt+h1113872 1113873 v(t x y) + 1113946

t+h

tvt s X

lowastZΓs Y

ZΓs1113872 1113873

+ G s XlowastZΓs Y

ZΓs Zs Γs1113872 1113873ds

(36)



G(t x y Z Γ) 0 (37)



v(t x y) ecfDtxy

minus cfcm( 1113857E(t) (38)




can get


lowastZΓt



(39)





Tcm( )( )|zt z1113876 1113877

inf(ZΓ)isinV(t)

E ecf X lowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

lowastZΓt

z

Dt

1113890 1113891

v tz

Dt

y1113888 1113889

(40)



minusμ + n

lowastZΓt 21113872 11138731113872 1113873 μ + n

lowastZΓt1113872 1113873

σ2Dt

Z

ΓVz

⎧⎨

⎩

minuscmθ n

lowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2σ2yDt

Z

ΓVy

+μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2D

2t

Z2

Γ2Vzz

+nlowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2cmθyDt( 1113857

2Z2

Γ2Vyy

+nlowastZΓt μ + n

lowastZΓt 21113872 11138731113872 1113873

2

σ2cmθyD

2t

Z2

Γ2Vzy

⎫⎪⎬

⎪⎭

(41)

V(T z y) ecfz


(42)




σ2DtMVz1113896

+(μ +(n2))

2

2σ2D

2t M

2Vzz minus

cmθn2(μ +(n2))

2σ2yDtMVy

+n2(μ +(n2))

2

2σ2cmθyDt( 1113857

2M

2Vyy

+n(μ +(n2))

2

σ2cmθyD

2t M

2Vzy1113897

(43)





yVy(t z y) minuscf

cm

V(t z y)

y2Vyy(t z y)

cf cf + cm1113872 1113873

c2m

V(t z y)


cm

V(t z y)

(44)




σ2DtMcf1113896

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

21113897E(t)V(t z y)

(45)










lowastt




ecfz


(46)

Here

B minusμ + n

lowast2( 1113857( 1113857 μ + nlowast

( 1113857

σ2DtMlowastt cf

+μ + n

lowast2( 1113857( 11138572

2σ2D

2t Mlowast 2t c

2f +

cfθnlowast2 μ + n

lowast2( 1113857( 1113857

2σ2DtMlowastt

+nlowast2 μ + n

lowast2( 1113857( 11138572

2σ2θ2D2


minusnlowast μ + n

lowast2( 1113857( 11138572

σ2c2fθD

2t Mlowast 2t

(47)

0 000

002

004

006

008

010

n


0ndash

000

002

004

006

008

010

n


0

000

002

004

006

008

010

n


0


000

002

004

006

008

010

n





ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)



dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b


Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt


Zlowast 2t

2Ylowast 2t

dt

(50)


dXlowastt rX



which implies

d lnYlowastt

Dtcmθnlowast2

blowastt


Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm


2minus12D



nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant


2minus12D



nlowast

+ μ( 1113857gt 0

(53)



lowastDt dX

lowastt minus rX



(54)

So








Corollary 1

Ylowasts Vm s X




Appendix

Define


σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)







In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References


























t1113864 1113865tge0) by


s

tH r Xr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdXr +

12

1113946s


(17)




Lemma 1 Define


nlowastZΓ

1113872 1113873

blowastYZΓ

t ZtΓtt1113882 1113883

tge0 nlowastYZΓ

t ZtΓtt1113882 1113883

tge01113874 1113875

(18)



lowastyzct and n

lowastyzct are





tain that

Yt Vm t x YZΓT1113872 1113873 (19)




s

tH r X

πr Y

ZΓr Zr Γr1113872 1113873dr

+ 1113946s

tZrdX

πr +

12

1113946s

tΓrdlangX

πrangr

(20)


deminus 1113946

r

tβ(u π)du

YZΓr e

minus 1113946r

tβ(u π)du

minus H r Xπr Y

ZΓr Zr Γr1113872 11138731113960

+ rXπr + b

πr μ + n

πr( 1113857( 1113857Zr

+bπ2r σ2


+ eminus 1113946

r

tβ(u π)du

σZrdW(r)

(21)


r




T

tβ(u π)du

YZΓT |X

πt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ J

πm t x Y

ZΓT1113872 1113873

(22)



1113872 1113873YZΓt dt + b

lowastYZΓt ZtΓt

t ZtσdWt (23)


deminus 1113946

r

tβ u πlowastZΓ

1113872 1113873duY

ZΓr e

minus 1113946r

tβ u πlowastZΓ


t ZtΓtt ZtσdWt

(24)


Yt E eminus 1113946

T

tβ u πlowastZΓ

1113872 1113873duY

ZΓT |XlowastZΓt x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

JπlowastZΓ

m t x YZΓT1113872 1113873ge J

πm t x Y

ZΓT1113872 1113873

(25)


Yt Vm t x YZΓT1113872 1113873 (26)








E ecf XlowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

πt x Y

ZΓt y1113876 1113877

(27)



v(t x y) (28)





lowastyzct n





F(t x y z b) +b2σ2

2c1113896 1113897 (29)

Here


minus Dt

cmθn2b

2y +(rx + b(μ + n))z1113896 1113897

(30)

Define


s

tF r Xr Y

Zr Zr1113872 1113873dr + 1113946

s

tZrdXr s isin [t T]

(31)




ZZb) isin Π





lowastyzbt are







G(t x y Z Γ) (32)

v(T x y) ecfx


(33)

where


lowastZΓt1113872 11138731113960 1113961vx

+σ2 blowastZΓt1113872 1113873

2

2vxx +


2

2blowastZΓt yvy

+σ2 blowastZΓt1113872 1113873

2

2Z2vyy + σ2 b

lowastZΓt1113872 1113873

2Zvxy

(34)




ZΓt+h1113872 1113873 (35)


s )



ZΓt+h1113872 1113873 v(t x y) + 1113946

t+h

tvt s X

lowastZΓs Y

ZΓs1113872 1113873

+ G s XlowastZΓs Y

ZΓs Zs Γs1113872 1113873ds

(36)



G(t x y Z Γ) 0 (37)



v(t x y) ecfDtxy

minus cfcm( 1113857E(t) (38)




can get


lowastZΓt



(39)





Tcm( )( )|zt z1113876 1113877

inf(ZΓ)isinV(t)

E ecf X lowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

lowastZΓt

z

Dt

1113890 1113891

v tz

Dt

y1113888 1113889

(40)



minusμ + n

lowastZΓt 21113872 11138731113872 1113873 μ + n

lowastZΓt1113872 1113873

σ2Dt

Z

ΓVz

⎧⎨

⎩

minuscmθ n

lowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2σ2yDt

Z

ΓVy

+μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2D

2t

Z2

Γ2Vzz

+nlowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2cmθyDt( 1113857

2Z2

Γ2Vyy

+nlowastZΓt μ + n

lowastZΓt 21113872 11138731113872 1113873

2

σ2cmθyD

2t

Z2

Γ2Vzy

⎫⎪⎬

⎪⎭

(41)

V(T z y) ecfz


(42)




σ2DtMVz1113896

+(μ +(n2))

2

2σ2D

2t M

2Vzz minus

cmθn2(μ +(n2))

2σ2yDtMVy

+n2(μ +(n2))

2

2σ2cmθyDt( 1113857

2M

2Vyy

+n(μ +(n2))

2

σ2cmθyD

2t M

2Vzy1113897

(43)





yVy(t z y) minuscf

cm

V(t z y)

y2Vyy(t z y)

cf cf + cm1113872 1113873

c2m

V(t z y)


cm

V(t z y)

(44)




σ2DtMcf1113896

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

21113897E(t)V(t z y)

(45)










lowastt




ecfz


(46)

Here

B minusμ + n

lowast2( 1113857( 1113857 μ + nlowast

( 1113857

σ2DtMlowastt cf

+μ + n

lowast2( 1113857( 11138572

2σ2D

2t Mlowast 2t c

2f +

cfθnlowast2 μ + n

lowast2( 1113857( 1113857

2σ2DtMlowastt

+nlowast2 μ + n

lowast2( 1113857( 11138572

2σ2θ2D2


minusnlowast μ + n

lowast2( 1113857( 11138572

σ2c2fθD

2t Mlowast 2t

(47)

0 000

002

004

006

008

010

n


0ndash

000

002

004

006

008

010

n


0

000

002

004

006

008

010

n


0


000

002

004

006

008

010

n





ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)



dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b


Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt


Zlowast 2t

2Ylowast 2t

dt

(50)


dXlowastt rX



which implies

d lnYlowastt

Dtcmθnlowast2

blowastt


Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm


2minus12D



nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant


2minus12D



nlowast

+ μ( 1113857gt 0

(53)



lowastDt dX

lowastt minus rX



(54)

So








Corollary 1

Ylowasts Vm s X




Appendix

Define


σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)







In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References



























E ecf XlowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

πt x Y

ZΓt y1113876 1113877

(27)



v(t x y) (28)





lowastyzct n





F(t x y z b) +b2σ2

2c1113896 1113897 (29)

Here


minus Dt

cmθn2b

2y +(rx + b(μ + n))z1113896 1113897

(30)

Define


s

tF r Xr Y

Zr Zr1113872 1113873dr + 1113946

s

tZrdXr s isin [t T]

(31)




ZZb) isin Π





lowastyzbt are







G(t x y Z Γ) (32)

v(T x y) ecfx


(33)

where


lowastZΓt1113872 11138731113960 1113961vx

+σ2 blowastZΓt1113872 1113873

2

2vxx +


2

2blowastZΓt yvy

+σ2 blowastZΓt1113872 1113873

2

2Z2vyy + σ2 b

lowastZΓt1113872 1113873

2Zvxy

(34)




ZΓt+h1113872 1113873 (35)


s )



ZΓt+h1113872 1113873 v(t x y) + 1113946

t+h

tvt s X

lowastZΓs Y

ZΓs1113872 1113873

+ G s XlowastZΓs Y

ZΓs Zs Γs1113872 1113873ds

(36)



G(t x y Z Γ) 0 (37)



v(t x y) ecfDtxy

minus cfcm( 1113857E(t) (38)




can get


lowastZΓt



(39)





Tcm( )( )|zt z1113876 1113877

inf(ZΓ)isinV(t)

E ecf X lowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

lowastZΓt

z

Dt

1113890 1113891

v tz

Dt

y1113888 1113889

(40)



minusμ + n

lowastZΓt 21113872 11138731113872 1113873 μ + n

lowastZΓt1113872 1113873

σ2Dt

Z

ΓVz

⎧⎨

⎩

minuscmθ n

lowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2σ2yDt

Z

ΓVy

+μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2D

2t

Z2

Γ2Vzz

+nlowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2cmθyDt( 1113857

2Z2

Γ2Vyy

+nlowastZΓt μ + n

lowastZΓt 21113872 11138731113872 1113873

2

σ2cmθyD

2t

Z2

Γ2Vzy

⎫⎪⎬

⎪⎭

(41)

V(T z y) ecfz


(42)




σ2DtMVz1113896

+(μ +(n2))

2

2σ2D

2t M

2Vzz minus

cmθn2(μ +(n2))

2σ2yDtMVy

+n2(μ +(n2))

2

2σ2cmθyDt( 1113857

2M

2Vyy

+n(μ +(n2))

2

σ2cmθyD

2t M

2Vzy1113897

(43)





yVy(t z y) minuscf

cm

V(t z y)

y2Vyy(t z y)

cf cf + cm1113872 1113873

c2m

V(t z y)


cm

V(t z y)

(44)




σ2DtMcf1113896

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

21113897E(t)V(t z y)

(45)










lowastt




ecfz


(46)

Here

B minusμ + n

lowast2( 1113857( 1113857 μ + nlowast

( 1113857

σ2DtMlowastt cf

+μ + n

lowast2( 1113857( 11138572

2σ2D

2t Mlowast 2t c

2f +

cfθnlowast2 μ + n

lowast2( 1113857( 1113857

2σ2DtMlowastt

+nlowast2 μ + n

lowast2( 1113857( 11138572

2σ2θ2D2


minusnlowast μ + n

lowast2( 1113857( 11138572

σ2c2fθD

2t Mlowast 2t

(47)

0 000

002

004

006

008

010

n


0ndash

000

002

004

006

008

010

n


0

000

002

004

006

008

010

n


0


000

002

004

006

008

010

n





ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)



dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b


Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt


Zlowast 2t

2Ylowast 2t

dt

(50)


dXlowastt rX



which implies

d lnYlowastt

Dtcmθnlowast2

blowastt


Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm


2minus12D



nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant


2minus12D



nlowast

+ μ( 1113857gt 0

(53)



lowastDt dX

lowastt minus rX



(54)

So








Corollary 1

Ylowasts Vm s X




Appendix

Define


σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)







In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References



























Tcm( )( )|zt z1113876 1113877

inf(ZΓ)isinV(t)

E ecf X lowastZΓ

Tminus lnYZΓ

Tcm( )( )|X

lowastZΓt

z

Dt

1113890 1113891

v tz

Dt

y1113888 1113889

(40)



minusμ + n

lowastZΓt 21113872 11138731113872 1113873 μ + n

lowastZΓt1113872 1113873

σ2Dt

Z

ΓVz

⎧⎨

⎩

minuscmθ n

lowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2σ2yDt

Z

ΓVy

+μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2D

2t

Z2

Γ2Vzz

+nlowastZΓt1113872 1113873

2μ + n

lowastZΓt 21113872 11138731113872 1113873

2

2σ2cmθyDt( 1113857

2Z2

Γ2Vyy

+nlowastZΓt μ + n

lowastZΓt 21113872 11138731113872 1113873

2

σ2cmθyD

2t

Z2

Γ2Vzy

⎫⎪⎬

⎪⎭

(41)

V(T z y) ecfz


(42)




σ2DtMVz1113896

+(μ +(n2))

2

2σ2D

2t M

2Vzz minus

cmθn2(μ +(n2))

2σ2yDtMVy

+n2(μ +(n2))

2

2σ2cmθyDt( 1113857

2M

2Vyy

+n(μ +(n2))

2

σ2cmθyD

2t M

2Vzy1113897

(43)





yVy(t z y) minuscf

cm

V(t z y)

y2Vyy(t z y)

cf cf + cm1113872 1113873

c2m

V(t z y)


cm

V(t z y)

(44)




σ2DtMcf1113896

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

21113897E(t)V(t z y)

(45)










lowastt




ecfz


(46)

Here

B minusμ + n

lowast2( 1113857( 1113857 μ + nlowast

( 1113857

σ2DtMlowastt cf

+μ + n

lowast2( 1113857( 11138572

2σ2D

2t Mlowast 2t c

2f +

cfθnlowast2 μ + n

lowast2( 1113857( 1113857

2σ2DtMlowastt

+nlowast2 μ + n

lowast2( 1113857( 11138572

2σ2θ2D2


minusnlowast μ + n

lowast2( 1113857( 11138572

σ2c2fθD

2t Mlowast 2t

(47)

0 000

002

004

006

008

010

n


0ndash

000

002

004

006

008

010

n


0

000

002

004

006

008

010

n


0


000

002

004

006

008

010

n





ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)



dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b


Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt


Zlowast 2t

2Ylowast 2t

dt

(50)


dXlowastt rX



which implies

d lnYlowastt

Dtcmθnlowast2

blowastt


Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm


2minus12D



nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant


2minus12D



nlowast

+ μ( 1113857gt 0

(53)



lowastDt dX

lowastt minus rX



(54)

So








Corollary 1

Ylowasts Vm s X




Appendix

Define


σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)







In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References


























lowastt




ecfz


(46)

Here

B minusμ + n

lowast2( 1113857( 1113857 μ + nlowast

( 1113857

σ2DtMlowastt cf

+μ + n

lowast2( 1113857( 11138572

2σ2D

2t Mlowast 2t c

2f +

cfθnlowast2 μ + n

lowast2( 1113857( 1113857

2σ2DtMlowastt

+nlowast2 μ + n

lowast2( 1113857( 11138572

2σ2θ2D2


minusnlowast μ + n

lowast2( 1113857( 11138572

σ2c2fθD

2t Mlowast 2t

(47)

0 000

002

004

006

008

010

n


0ndash

000

002

004

006

008

010

n


0

000

002

004

006

008

010

n


0


000

002

004

006

008

010

n





ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)



dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b


Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt


Zlowast 2t

2Ylowast 2t

dt

(50)


dXlowastt rX



which implies

d lnYlowastt

Dtcmθnlowast2

blowastt


Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm


2minus12D



nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant


2minus12D



nlowast

+ μ( 1113857gt 0

(53)



lowastDt dX

lowastt minus rX



(54)

So








Corollary 1

Ylowasts Vm s X




Appendix

Define


σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)







In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References


























ecfDtxy

minus cfcm( 1113857

Vf(t x) v t x ecmR

1113872 1113873 eB(Tminus t)

ecf Dtxminus R( )

(48)



dYlowastt

Dtcmθnlowast2

blowastt Ylowastt

2dt + b


Ylowastt e

cmR

(49)


d lnYlowastt

Dtcmθnlowast2

blowastt

2dt + b

lowastt

Zlowastt

Ylowastt


Zlowast 2t

2Ylowast 2t

dt

(50)


dXlowastt rX



which implies

d lnYlowastt

Dtcmθnlowast2

blowastt


Zlowast 2t

2Ylowast 2t

1113888 1113889dt +Zlowastt

Ylowastt

dXlowastt minus rX


+ μ( 1113857( 1113857dt( 1113857

cm


2minus12D



nlowast

+ μ( 11138571113888 1113889dt

+ cmθnlowastDt dX

lowastt minus rX

lowastt dt( 1113857

(52)

Define constant


2minus12D



nlowast

+ μ( 1113857gt 0

(53)



lowastDt dX

lowastt minus rX



(54)

So








Corollary 1

Ylowasts Vm s X




Appendix

Define


σ2DtMcf

+(μ +(n2))

2

2σ2D

2t M

2c2f +

cfθn2(μ +(n2))

2σ2DtM

+n2(μ +(n2))

2

2σ2θ2D2

t M2cf cf + cm1113872 1113873

minusn(μ +(n2))

2

σ2c2fθD

2t M

2

(A1)







In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References




























In(n M t) minusDtcf

σ232μ + n1113874 1113875M

+Dtcfθ

2σ232n2

+ 2μn1113874 1113875M +D

2t c

2f

2σ2μ +

n

21113874 1113875M

2

+θ2D2

t cf cf + cm1113872 1113873

2σ2n3

+ 3μn2

+ 2μ2n1113872 1113873M2

minusc2fθD

2t

σ23n

2

4+ 2μn + μ21113888 1113889M

2 0

(A2)

IM(n M t) minusDtcf

σ2(μ + n) μ +

n

21113874 1113875

+Dtcfθ

2σ2n2 μ +

n

21113874 1113875 +

D2t c

2f

σ2μ +

n

21113874 1113875

2M

+θ2D2

t cf cf + cm1113872 1113873

σ2n2 μ +

n

21113874 1113875

2M

minus2c

2fθD

2t

σ2n μ +

n

21113874 1113875

2M 0

(A3)



21tθ21113872 1113873 +(μ2) minus θn

21t41113872 1113873 minus θμn1t


(A4)



2θ21113872 1113873

n1t2( 1113857 + μ( 1113857 2n1tθcf + n1t1113872 1113873

2θ2 cm + cf1113872 1113873 + cf1113876 1113877

(A5)



2θ2

1113888 1113889 minuscf

2+ cfθμ minus

cfθn1t

2+ θ2μ cf + cm1113872 1113873n1t1113890 1113891

+θn

21t

4+ θμn1t minus

μ2

1113888 1113889 2n1tθcf + n1t( 11138572θ2 cm + cf1113872 1113873 + cf1113960 1113961 0

(A6)






lowastt are all



Data Availability




Acknowledgments


References











































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ThePrinciple-AgentConflictProbleminaContinuous-Time