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Information and option pricings

Xin Guo

IBM T. J. Watson Research Center, P. O. Box 218, Yorktown,

NY 10598, USA

Abstract

How can one relate stock fluctuations and information-based human ac-

tivities? We present a model of an incomplete market by adjoining the

Black-Scholes exponential Brownian motion model for stock fluctuations

with a hidden Markov process, which represents the state of information in

the investors community. The drift and volatility parameters take differ-

ent values depending on the state of this hidden Markov process. Standard

option pricing procedure under this model becomes problematic. Yet, with

an additional economic assumption, we provide an explicit closed-form for-

mula for the arbitrage-free price of the European call option. Our model can

be discretized via a Skorohod imbedding technique. We conclude with an

example of a simulation of IBM stock, which shows that, not surprisingly,

information does affect the market.

AMS classification: 60J65; 60J10;

Tel: (914) 945 2348; Fax: (914) 945 3434; E-mail: xinguo@us.ibm.com.

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Keywords: Black-Scholes; Hidden Markov processes; Inside information; Ar-

bitrage; Equivalent martingale measure

1 Motivation

We begin our discussion with the classic case of Bre-X, a Canadian gold miningcompany. The mineral company stumbled on what looked like a huge gold cache

in Indonesia. Consequently, the stock price sky-rocketed for a while. Then, all

of a sudden, the volatility increased by orders of magnitude due to heavy inside

tradings. The reason turned out to be that a privileged few were aware of the

fraudulent gold assays performed by the company. The honeymoon was over and

the stocks crashed when this news became public (cf. Figure 1, New York Times,

May 5, 1997). Bre-X perished.

Figure 1: The rise and fall of Bre-X. (Source: New York Times, May 5, 1997.)

One of the morals of the story is: volatility increased when there was incom-

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plete informationsome people knew that the assays were fraudulent while the

rest did not. In general, it is reasonable to think that the existence of a one-sided

group of insiders (in this case short-sellers) will drive the market faster since they

will always be ready to sell if a buyer appears. They know something (or believe

they know something) that they think is worth money because others do not know

what they know (or believe they know).This story well-exemplifies the role of distribution of information in the in-

vestors community. Information is rarely, if ever shared, simultaneously by ev-

eryone. It is exactly this time difference that may create an arbitrage opportunity,

which never exists long. It appears, but is removed immediately once everyone

gets the information. And this information (or the lack of it) drives human

activity in the stock market.

With the view of understanding the stock market better, the obvious question

arises: Can and how does one link market movement and human activity? This

would tantamount to modeling stock fluctuations with information. This is the

theme of our paper.

A guided road map. Section 2 details the hidden Markov process that models

stock fluctuations and information changes. Section 3 shows the option pricing

scheme for our model. Section 4 outlines one possible discretization of our model.

Section 5 discusses why our model is fundamentally different from other models,

such as stochastic volatility models. Finally, Section 6 provides some preliminary

empirical evidence.

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2 A hidden Markov model with information

We incorporate the existence of inside information by modeling the fluctuations

of a single stock price Xt using an equation of the form

dXt = Xt( t) dt + Xt( t) dWt (1)

where ( t) is an additional stochastic process representing the state of information

in the investor community. ( t) is independent of Wt, the Weiner process. For

each state i, there is a known drift parameter i and a known volatility parameter

i. h ( t) ; ( t) i take different values when ( t) is in different states.

We assume that = ( t) is a Markov process which moves among a few (say,

2 or 3) states. ( t) = 0 at those times t at which the price change is not abnormal

and people believe that they are all well-informed in a seemingly complete mar-

ket; in this state ( t) = 0 ; ( t) = 0. But the process ( t) may take other values

than zero. ( t) = 1 when there are wild fluctuations in the stock price and peo-

ple suspect that some individuals or groups have extra information which is not

circulating among the mass of investors and thus would possibly bring wilder fluc-

tuations depending on the reaction of the investors. Here, ( t) = 1 ; ( t) = 1.

1 may be larger or smaller than 0 depending on the nature of inside information,

therefore this state may divide into two extra states where informed investors be-lieve the company will prosper or decline. Furthermore, some inside groups may

actually be misled and the model could include a state which would indicate that

there is a group of investors who erroneously believe that the companys fortunes

are going to change for the positive, and another state for the negative. More

generally, one can use the state space S = f 0 ; 1; 2; : : : ; Ng for ( t) to model more

complex information structures.

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If we assume that the s are distinct then it is no loss of generality to assume

that ( t) is actually observable, since the local quadratic variation of Xt in any

small interval to the left of t will yield ( t) exactly. (For details, see McKean,

1969.) Hence, even ifX( t) is not Markovian, h X( t) ; ( t) i is jointly so.

It is conceivable that sometimes insiders will try to manipulate their buying

and selling in such a way that the existence of such information is not detectablefrom the change of volatilities, namely s are identical. The problem of de-

tecting the state change of ( t) when s remain unchanged appears to be hard

to solve mathematically. It is plausible that change in information distribution,

hence predictability, manifests itself in the diffusion coefficient in the form of

both stochastic volatility and drift.

A two-state model. For ease of exposition, we focus on the two-state case, inwhich ( t) alternates between 0 and 1 such that

( t) =

8

>

:

0 ; when the market seems complete, and

1 ; when some people have (or believe they have) inside information;

(2)

where 0 6= 1.

Suppose, further, that each piece of information flow is a random process Yi,

and Y1;

Y2; : : : ;

Yn being i.i.d processes, then their super-imposed process (underminor technical restrictions) is Poisson. Therefore, let i denote the rate of leaving

state i, i the time of leaving state i, then

P( i > t) = e it

; i = 0 ; 1 (3)

The memoryless property of this process is plausible in that, from a practical

standpoint, the information flow be identified more easily otherwise.

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3 Option pricings and arbitrage

3.1 Completing the marketnew securities COS

It is easy to see that the model is not complete, according to Harrison and Pliska

(1981), Harrison and Kreps (1979), because of the additional process ( t) . In

other words, ( t) is a bounded adapted process with respect to the -algebra Ft

generated by Xt (denoted as FX), but is not adapted to the -algebra generated by

Wt (written as FW).

One way (by D. Duffie) to complete the market is as follows: at each time t,

there is a market for a security that pays one unit of account (say, a dollar) at the

next time ( t) = inff u > t ( u ) 6= ( t) g that the Markov chain ( t) changes state.

That contract then becomes worthless (i.e., has no future dividends), and a new

contract is issued that pays at the next change of state, and so on. Under natural

pricing, this will complete the market, and provide unique arbitrage-free prices to

the hedge options on the underlying risk asset. (For reference, see Harrison and

Pliska, 1981).

One can think of this as an insurance contract that compensates its holder for

any losses that occur when the next state change occurs. Of course, if one wants

to hedge a given deterministic loss C at the next state change, one holds C of the

current change-of-state (COS) contracts.

3.2 Pricing and no arbitrage

As an assumption analogous to the assumption of the pricing of the underlying

risky stock, it is natural to propose that the current COS contract trade for a

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price of

V( t) = E

e r+ k( ( t) ) ] ( ( t) t)

Ft

; (4)

where k: f 0; 1 g ! is given, and can be thought of as a risk-premium coefficient.

More precisely, the current COS contract price is

V(

t) =

J(

(

t) ) ;

(5)

J( i) =( i)

r+ k( i ) + ( i ); (6)

recalling that ( ( t) ) is the intensity of the point process N that counts changes of

state.

One the other hand, It can be shown (Harrison and Kreps (1979), Harrison

and Pliska (1981)) that the absence of arbitrage is effectively the same as the

existence of a probability measure Q, equivalent to P, under which the price of

any derivative is the expected discounted value of its future cash flow.

Given such a measure Q, we must therefore have

V( t) = EQ

e r( ( t) t)

Ft

;

(7)

where EQ denotes expectation under Q. The price of the current COS security is

of course zero after the next change of state.

Under Q, the counting process N has intensity of the form Q ( ( t) ) , Solving

the expression, we get V( t) = JQ ( ( t) ) , where

JQ ( i) =Q ( i)

r+ Q ( i)(8)

Of course, J = JQ, and therefore

Q ( i ) =r( i)

r+ k( i )(9)

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The same exercise applied to the underlying risky-asset implies that its price pro-

cess S must have the form

dS ( t) = ( r d( t) ) S ( t) dt + St( t) dBQ

; (10)

where BQ is a standard Brownian motion under Q.

Now the usual techniques from Harrison and Kreps (1979) and Harrison and

Pliska (1981) can be applied to get complete market and unique pricing for any

derivatives with appropriate square-integrable cash flows.

Theorem 1 Given Eq. (10), COS, and a riskless interest rate r, the arbitrage free

price of a European call option with expiration date T and strike price K is:

Vi ( T; K; r) = EQ e rT

( XT K)+ ( 0 ) = i (11)

= e

rT

Z

0

Z T

0y( ln ( y + K) ; m( t) ; v( t) ) fi ( t; T) dtdy; (12)

where ( x; m( t) ; v( t) ) is the normal density function with expectation m ( t) and

variance v( t) , and

f0 ( t; T) = e 1Te( 1 0 ) t

T t

01t 1 = 2J

1 2 ( 01T t + 01T2

)

1= 2

+ 0J0 2 ( 01T t + 01T2

)

1=

2

; (13)

f1(

t;

T) =

e 0T

e( 0 1 ) t

T t

01 t1

=

2

J 1

2(

01T t+

01T2

)

1=

2

+ 1J0 2 ( 01T t + 01T2

)

1=

2

; (14)

m( t) = ( d1 d0 1 = 2 ( 20

21 ) ) t + ( r d1 1= 2

21 ) T; (15)

v( t) = ( 20 21 ) t +

21T; (16)

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where Ja ( z) is the Bessel function such that (cf. Oberhettinger and Badii, 1973)

Ja ( z) =

1

2z

a

n= 0

(

1)

n( z= 2 ) 2n

n!( a + n + 1 ); (17)

Ya( z) = cot( a ) Ja ( z) csc ( a) J a ( z) (18)

In particular, when 0 = 1 ; 1 = 0, we have fi ( t; T) = t T, and therefore above

the equations reduce to the classical Black-Scholes formula for European options.

The key idea is to calculate the probability distribution function of a telegraph

process. This was obtained independently and earlier by Di Masi et. al. (1994).

Our Laplace transform based approach is entierly different. The details of our

proof are given in Appendix A.

Comparing Eq. (10) with a geometric Brownian motion process with drift r

and variance , d( t) is of special interest to us. A careful examination reveals

that this very extra term d( t) differentiates our model from the standard stochastic

volatility and Markov volatility models, in that it invalidates the martingale pricing

approach. Moreover, it provides us a way to understand the flow of the infor-

mation. The drift differs from the riskless interest rate r by d1 d0 when there is

some information flow and hence the arbitrage opportunity emerges. It also sug-

gests the difference between the case of pure noise (i.e., 0 6= 1 ; d0 = d1) and

the case when there may exist an inside information (i.e., 0

6= 1

; d1

6= d0

).

4 A discretization of the CRR type

To facilitate numerical simulations, we present one way of discretizing our con-

tinuous market model along the vein of Cox, Ross, and Rubinstein (Cox, Ross,

and Rubinstein, 1979). It is worth pointing out that this methodology applies also

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to the general case where the the hidden Markov process ( t) takes more than

two states, i.e., the state space can be S = f 0 ; 1; : : : ; Ng , when, for example, more

complex information patterns can be imposed.

Suppose the time interval 0; t is divided into n sub-intervals such that t = nh.

Let X = ( Xk) where Sk is a price at time kh, and define:

Xkk

= ( Xk; k) = ( X( kh ) ; ( kh ) ) ; (19)

then the following recurrence is obtained,

( Xn ; ( n ) ) = ( ( n ) ( n 1 ) )n ( Xn 1 ; ( n 1) ) ; (20)

where i jn are i.i.d random variables taking values uj with probability pj ( i 1 j +

( 1 ) i 1 j e ih ) and 1= uj with probability ( 1 pj ) ( i 1 j + ( 1)i 1 j e ih ) re-

spectively (i;

j=

0;

1), where

ai = e ih

; ui = ei

p

h; pi =

ih + ip

h 0 52i h

2ip

h(21)

By the memoryless property of i ; ( Xkk

) ; i = 0 ; 1 is a Markov chain. More pre-

cisely, the Markov chain f Xg = ( Xnn ) with initial state X0 = x is a random walk

on the set Ex = f xur r = 0m + 1n ; m; n 2 Z; u = e

p

hg .

Intuitively Eq. (20) provides the right discretization and indeed it can be

proved.

Theorem 2 Xkk

converges in distribution to Xt as given in Eq. (1) when h ! 0.

To prove that Xn ! X in distribution, it is equivalent to show the convergence

E f( Xn ) ! E f( X) for each bounded and continuous function f( ) . Using standard

techniques (cf. Billingsley, 1968; Kurtz, 1985; Kushner and Huang, 1984; Skoro-

hod, 1956), we define new processes Xn and X, the piecewise linear interpolation

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ofXn and X. Via a Skorohod imbedding technique, we prove that Xn converges to

X with probability one, hence the convergence in distribution of Xn to X.

Proof Sketch: [of Theorem 2] The key here is to calculate the characteristic func-

tions ofYt = lnXt and Y( n )

n , (cf. Feller, 1971) where

Yt = Y0 +

Z t

0(

1

22 ) ds +

Z t

0dWs ; (22)

Y( n)n = Y

( n 1 )n 1 j

p

h ; ( j = 0; 1) (23)

Without loss of generality, let Y( 0 )

0 = 0, let

fj ( t) = E E eicYt ( 0) = j

= E eR t

0 ic ic12

2

12 c

22ds ( 0) = j ; (24)

then starting from time 0, between time 0 and h, either (

h) 6=

(

0) =

j with prob-ability 1 e jh, or ( h) = ( 0 ) = j with probability e jh and the process starts

afresh because of the memoryless property of the exponential function. Therefore

we have

fj ( t) = e jheh ( icj ic

12

2j

12 c

22j ) fj ( t h) + jh f1 j ( t) + o ( h ) (25)

By Taylors expansion, we have

fj ( t) = 1 + ( icj ic122j 12

c22j j ) h + o ( h) fj ( t) f0

j ( t) h + o ( h )

+ jh f1

j ( t) ; (26)

thus f0 and f1 satisfy first order system of ODEs

8

>

:

f00 ( t) = ( ic0 ic12

20

12 c

220 0 ) f0 ( t) + 0f1 ( t)

f01 ( t) = ( ic1 ic1221

12 c

221 1 ) f1 ( t) + 1f0 ( t) ;

(27)

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with initial conditions fj ( 0) = 1; f0

j ( 0) = icj 12 ic

2j

12 c

22j ; ( j = 0 ; 1 ) . This

system is stable and has a unique solution for any fixed c.

On the other hand, let Ynn = lnXnn as defined in Eq. (23), and let

fj k = E E eicYk ( 0 ) = j ; ( j = 0 ; 1 ) ; 0 k n (28)

Then, we have

f0 k = E E eicY

kk ( 0) = 0

= E ( eicY( k 1

k 1+ ic0

p

hp0 + eicY

( k 1k 1

ic0p

h( 1 p0 ) ) e

0h ( 1 ) = 0

+ ( 1 e 0h ) ( eicY

( k 1k 1

+ ic1p

hp1 + eicY

( 1

k 1 ic1

p

h( 1 p1 ) ) ( 1 ) = 1

= f0 k 1 eic0

p

h 0hp0 + ( 1 p0 ) e

ic0p

h 0h

+ 0h f1 k 1 eic1

p

hp1 + e ic1

p

h( 1 p1 )

=

f0 k 1

eic0

p

h 0h

p0+ (

1

p0)

e ic0

p

h 0h+

0h f1 k 1

= f0 k 1 f0 n 1h0 1= 2f0 n 1c220h + ( 0 1= 2

20 ) ich f0 k 1

+ 0h f1 k 1 + o( h) (29)

By linear interpolation, it is not hard to see that when h ! 0, there exists fj

which satisfies

f00 ( t) = ( ic0 1

2ic20

1

2c220 0 ) f0 ( t) + 0f1 ( t) (30)

Similarly, we have

f01 ( t) = ( ic1 1

2ic21

1

2c221 1 ) f1 ( t) + 1f0 ( t) (31)

and initial conditions fj ( 0 ) = 1 ; f0

j ( 0 ) = icj 12 ic

2j

12 c

22j. Therefore, by

the uniqueness of the solution to the ODE Eq. (27), fi = fi . Hence Theorem 2

follows immediately.

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5 Our model vs. other models

There has been extensive work on modeling stock fluctuations with stochastic

volatility (cf. Anderson, 1996; Hull and White, 1987; Stein, 1991; Wiggins,

1987), Markov volatility (Di Masi et. al. 1994), and uncertain volatility (Avel-

laneda, Levy, and Paras, 1995). Many efforts have been made to study financial

markets with different information levels among investors (cf. Duffie and Huang,

1986; Ross, 1989; Anderson, 1996; Karatzas and Pikovsky, 1996; Guilaume et.

al., 1997; Grorud and Pontier, 1998; Imkeller and Weisz, 1999). Especially, Kyle

(1985) considered a dynamic model of inside trading with sequential auctions, in

which the informational content of prices and the values of private information

to an insider are examined. Lo and Wang (1993) used Ornstein-Uhlenbeck (O-

U) processes in their adjustment to the Black-Scholes model (Black and Scholes,

1973; Merton 1973) to induce the drift term via option formula.

Our model vs. stochastic volatility. It is worth pointing out that the model

we propose here is fundamentally different from various models with stochastic

volatility or uncertain volatility. This is because the drift ( t) is also driven by the

hidden Markov process ( t) , which, in consequence, changes the option pricing

methodology.

Our model vs. Markov volatility. The work of Di Masi et. al. (1994) on

Markov volatility emphasizes more on aspects of hedging strategies than of op-

tion pricing issues. Moreover, they seemed to have overlooked the fact that the

martingale pricing approach would not be applicable for the general case when

the drift term is non-zero, therefore their pricing procedure would be flawed.

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Our model vs. O-U processes. O-U processes have the nice property that when

the price goes too negative, the drift term will pull it back, which makes eco-

nomic sense. It is however known to probabilists that O-U process is a rescaled

Brownian motion; therefore it does have its limitations to adjusting the Black-

Scholes model. Moreover, for tractability reasons, the model proposed by Lo and

Wang (1993) assumes a fixed volatility.In our model, the information component is reflected in both the drift and

volatility term. Therefore, not surprisingly, the option pricing formula is function-

ally dependent on the drift. Moreover, unlike the trending O-U processes which

are Markovian, in our model while X( t) is not Markovian, h X( t) ; ( t) i is jointly

so. It provides a simple and feasible way to connect historical data and current sit-

uation. Furthermore, it captures our earlier intuition that the market never exists

independently of the information distribution.

Remark. An anticipated criticism comes from the usage of the term inside in-

formation, perhaps due to its not-so-glorious image in our (hopefully) efficient

market. It is, therefore, worth pointing out that inside information is merely a

convenient way to describe the hidden Markov process ( t) , which is driven by

some market force, and can be interpreted in a broader way to reflect the noise

exemplified by d( t) , that goes beyond the Black-Scholes and other standard mod-

els.

For pricings of other types of hedge options such as perpetual lookback op-

tions, Russian options, perpetual American options, interested readers are referred

to (Guo, 1999).

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6 Empirical results and conclusions

We conclude our paper with an example borrowed from the ongoing empirical

study with Ed. Pednault at IBM (Guo and Pednault, 2000). Figure 6 illustrates

our notion of information structure that is present in the IBM stock price.

Figure 2: IBM stock price.

There are many immediate questions that spring to our mind. We recognize

that it is not completely realistic to assume that 0 6= 1. We believe, however,

the assumption that information structure is closely related to fluctuations in drift

and volatilities stands to reason, while the mathematical tractability is retained. It

is also far from clear how drift and volatility are intrinsically related with respect

to change in information distribution. This question requires further investigation

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and statisticians and experimental economists might be able to provide possible

answers. There are also cases in which the emergence of inside information will

have a certain delay time, such that ( t) and ( t) will be generated by different

processes ( t) and 0 ( t) . It will be interesting to study these models. Furthermore,

another direction is to model the actions investors undertake upon receiving new

information.It is worth pointing out that our option pricing approach relies heavily on a

rather strong economic assumption: the existence of a COS contract. It is not clear

how feasible this assumption is. This brings up a even more basic question: is

there a better way to incorporate information distribution into stock fluctuations?

Indeed, a simple model like ours, whose primary goal is to capture the real-

ity in stock market without sacrificing tractability, has already gone beyond the

boundary of the general framework of martingale approach. Therefore, we feel

that our model may serve as little acorn from which great oaks could grow.

Acknowledgements

This model was jointly developed with Larry Shepp. He deserves special thanks

for motivating me to pursue it further. Darrell Duffie, Dan Ocone, and Michael

Harrison made valuable suggestions to improving both the content and presenta-

tion of this paper. Ed. Pednault provided me with Figure 6. Part of this work was

supported by DIMACS. I thank the hospitality of the University of California at

Berkeley.

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Stein, E. M., and Stein, C. J., 1991, Stock prices distribution with stochastic volatility, an

analytic approach, Review of Financial Studies 4, 727752.

Wiggins, J. B., 1987, Option values under stochastic volatility. Theory and empirical

evidence, Journal of Financial Economics 19, 351372.

A Proof of Theorem 1

Since the arbitrage price of the European option is the discounted expected value

ofXt under the equivalent martingale measure Q, we have

Vi ( T; K; r) = EQ e rT

( XT K)+ ( 0) = i (32)

Recalling that

Xt = X( 0 ) exp(

Z t

0( r d( s ) 1= 2

2( s ) ) ds +

Z t

0( s ) dW

s ) ; (33)

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the key point is to calculate the instantaneous distribution of X( T) . Now let Yt =

lnXt, then

Yt = Y( 0 ) +

Z t

0( r d( s ) 1= 2

2( s ) ) ds +

Z t

0( s ) dW

s (34)

If we consider the probability distribution function fi ( t; T) , where fi ( t; T) is the

probability distribution function of Ti, (Ti being the total time between 0 and T

during which ( t) = 0, starting from state i), then by the well-known property of

conditional expectations, we have

Vi ( T; K; r) = E e rT

( XT K)+ ( 0) = i

= e rTE E ( XT K)+ Ti ( 0) = i

= e rTEi E ( XT K)+ Ti F ( 0) = i

=

e rT

Z

0

Z T

0 y(

ln(

y+

K)

x;

m(

t) ;

v(

t) )

fi(

t;

T)

dydt;

(35)

where x = X( 0) , ( x; m( t) ; v ( t) ) is the normal density function with expectation

m( t) and variance v( t) .

Clearly, we have:

m( t) = ( d1 d0 1 = 2 ( 20

21 ) ) t + ( r d1 1= 2

21 ) T; (36)

v( t) = ( 20 21 ) t +

21T; (37)

and

( x; m( t) ; v( t) ) =1

p

2v( t)exp(

( x m( t) ) 2

2v( t)) (38)

Now the key is to calculate fi ( t; T) . Notice that

fi ( t; T) dt = P(

Z T

0

0( s ) ds 2 dt) ; (39)

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and let

i ( r; T) = E e r

R T0 0 ( s ) ds ( 0 ) = i

=

Z

0e rtfi ( t; T) dt

= Lr( fi ( ; T) ) ; (40)

then we have

i ( r; T) = e

rTe iTi 0 + e iT1 i 0

+

Z T

0e iui1

i ( T u) e rui 0 du ; (41)

i.e.,

0 ( r; T) = e rTe 0T +

Z T

0e 0u01 ( T u ) e

rudu ; (42)

1(

r;

T) =

e 1T

+

Z T

0 e 1u

10(

T

u)

du (43)

Taking Laplace transforms on both sides, and writing

Ls ( i ( r; ) ) = Ls Lr( fi ( ; T) ) ( r; )

=

Z

0e sTi ( r; T) dT

= i ( r; s) ; (44)

then

0 ( r; s) =1

r+ s + 0+

0r+ s + 0

1 ( r; s) (45)

1 ( r; s) =1

s + 1+

1s + 1

0 ( r; s) (46)

Solving these equations, we obtain:

0 ( r; s) =s + 0 + 1

s2 + s1 + s0 + rs + r1(47)

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Taking the inverse Laplace transform on Eq. (47) with respect to ryields

L 1r ( 0 ( r; s) ) ( w; ) = ( 1 +0

s + 1) exp(

s( s + 0 + 1 )

s + 1w)

= exp( sw s0

s + 1w)

+

0s + 1

exp( sw s0

s + 1w) (48)

By a well-known property of the Laplace transform, effectively we can first take

the Laplace inverse transform of the above formula with respect to s (the conver-

gence of the integrand is obvious) and obtain

L 1s ( L 1r ( 0 ( r; s) ) ) ( w; ) ( ; v)

= L 1s (s + 0 + 1

s + 1exp(

s( s + 0 + 1 )

s + 1w) ) ( ; v)

= L 1

s

( exp( s( s + 0 + 1 )

s + 1w)

+

0s + 1

exp(

s( s + 0 + 1 )

s + 1w) ) ( ; v)

= e 1ve( 1 0 ) wL 1s ( e ws +

01ws

+

0s

e ws +01w

s) ( ; v) (49)

Recall that

L 1s ( e

as) ( v) = ( v a) ; (50)

and that if

L 1s ( g( s) ) ( v) = f( v) ; (51)

then

L 1s ( s 2c 1g ( s + b = s) ) ( v) =

Z v

0f( u) ( v u ) = ( bu) cJ2c 2( buv bu

2)

1 = 2 du ;

(52)

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where Ja ( z) s are Bessel functions as given in Eqs. (17), and (18).

Therefore we have (for w v):

L 1s (s + 0 + 1

s + 0exp(

s( s + 0 + 1 )

s + 1w) ) ( ; v)

= e 1ve( 1 0 ) w (

Z v

0( u w)

( v u)

01u 1 = 2J

1 2 ( 01uv + 01u2

)

1= 2 du

+ 0

Z v

0( u w) J0 2( 01uv + 01u

2)

1=

2 du )

= e 1ve( 1 0 ) w ( v w

01w 1= 2J

1 2( 01wv + 01w2

)

1= 2

+ 0J0 2 ( 01wv + 01w2

)

1=

2) (53)

Thus we obtain f0 ( w; v) , the distribution function ofT0, such that

f0 ( w; v) = L

1s ( L

1r ( 0 ( r; s) ( w; ) ) ( ; v) )

= e 1ve( 1 0 ) w ( v w

01w1 = 2J

1 2( 01wv + 01w2

)

1 = 2

+ 0J0 2 ( 01wv + 01w2

)

1=

2) (54)

Similarly we have:

f1 ( w; v) = e 0ve( 0 1 ) (

v w

01w21

=

2J

1 2 ( 01wv + 01w2

)

1=

2

+ 1J0 2 ( 01wv + 01w2

)

1 = 2) ;

(55)

Now the theorem is immediate.

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