value function and optimal rule on the optimal stopping...

Research ArticleValue Function and Optimal Rule on the Optimal StoppingProblem for Continuous-Time Markov Processes

Lu Ye

College of Economics and Management and Zhejiang Provincial Research Center for Ecological CivilizationZhejiang Sci-Tech University Hangzhou 310018 China

Correspondence should be addressed to Lu Ye zjwzajylhotmailcom

Received 30 May 2017 Accepted 5 September 2017 Published 9 October 2017

Academic Editor Antonio Di Crescenzo

Copyright copy 2017 Lu YeThis is an open access article distributed under the Creative Commons Attribution License which permitsunrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers the optimal stopping problem for continuous-timeMarkov processesWe describe themethodology and solvethe optimal stopping problem for a broad class of reward functions Moreover we illustrate the outcomes by some typical Markovprocesses including diffusion and Levy processes with jumps For each of the processes the explicit formula for value function andoptimal stopping time is derived Furthermore we relate the derived optimal rules to some other optimal problems

1 Introduction

Let (ΩF 119875) be a complete probability space the problemstudied in this paper is to find the optimum

V (119909) = sup120591

E119909 [eminus119903(120591minus119905)119891 (119883120591) | F119905] (1)

where 119903 gt 0 is the discount rate and 120591 ge 119905 is a stoppingtime The process 119883119905 is a continuous-time Markov processwith starting state 119883119905 = 119909 When 119883119905 denotes the stockprice and 119891(sdot) is the payoff function V(119909) is the pricingexpression for American option (eg see Wong [1]) When119891(sdot) is an investorrsquos utility function about the stock price 119883119905the optimum solution indicates the best time to buy or sell thestock (eg see McDonald and Siegel [2])

Because of the Markov property the value function V(119909)can be written as

V (119909) = sup120591

E119909 [eminus119903(120591minus119905)119891 (119883120591) | 119883119905 = 119909]≜ sup120591

E119909 [eminus119903(120591minus119905)119891 (119883120591)] (2)

where E119909(sdot) denotes the conditional expectation E119909(sdot | 119883119905 =119909) The function 119891(sdot) is called the reward function Thepossibility that 120591 = infin is allowed andmaking the conventionthat

eminus119903120591119891 (119883120591) = 0 if 120591 = infin (3)

it means that if an option is never exercised then its rewardpayment is valueless for the investor

In the paper we are interested in determining both anoptimal stopping time 120591lowast and value function V(119909) for a largeclass of reward functions We show the conditions for rewardfunctions and deduce the explicit optimal rules for generalcontinuous-time Markov processes including diffusion andLevy processes with jumps

The study of optimal stopping time for stochastic pro-cesses especially geometric Brownian motion has a longhistory in finance literature Under the assumption that119883119905 isgeometric Brownianmotion the seminal paper byMcDonaldand Siegel [2] puts forward the problem with the rewardfunction 119891(119909) = 1 minus 119909 as a model to illustrate the financialdecision making Hu and Oslashksendal [3] solved the problemin multidimensional cases when 119891(119909) = sum119899119894=1 119909119894 minus sum119898119894=119899+1 119909119894but they restricted the stopping time 120591 in a bounded intervalRecently Nishide and Rogers [4] extended the problem byrelaxing the restriction on the stopping time For the otherforms of value functions Pedersen and Peskir [5] solved theproblem by taking some special diffusion processes

The purpose of our work is threefold Firstly we intendto make clear the assumptions on reward function 119891(119909) insuch a way that the explicit value can be generalized to alarger class of issues For this purpose throughout this articlewe assume that the function 119891(119909) is nonincreasing concaveand twice continuous differentiableThese properties are very

HindawiChinese Journal of MathematicsVolume 2017 Article ID 3596037 10 pageshttpsdoiorg10115520173596037

2 Chinese Journal of Mathematics

powerful in the following proof as we will see Secondlywe pay attention to general Markov processes includingdiffusion and Levy processes with jumpsWith the help of theinfinitesimal generator we obtain an explicit formula for thevalue function and the stopping timeThirdly we find that theoptimal problem (2) is equivalent to other optimal problemslike

119908 (119909) = sup120591

E119909 [int120591119905eminus119903(119904minus119905)119891 (119883119904) d119904] (4)

119908 (119909) = inf120591E119909 [int120591119905eminus119903(119904minus119905)119891 (119883119904) d119904] (5)

This work is inspired by Pedersen and Peskir [5] who verifiedthe equivalence of problems (2) and (4) for 119891(119909) = 119909 where119883119905 is an Ornstein-Uhlenbeck process Note that we do notconsider the case 119891(119909) = 119909 so our approach to deal with theoptimal problem is different from that of Pedersen and PeskirMoreover our work naturally explore the explicit solutionsof the new optimal problem (4) for a larger class of rewardfunctions and underlying processes119883119905

The paper is organized as follows In Section 2 theexplicit value function and optimal stopping time are derivedfor a general Markov process along with the condition forthe reward functions Section 3 discusses some applicationsto diffusion which include Brownian motion with driftgeometric Brownian motion and the Ornstein-Uhlenbeckprocess Section 4 displays some concrete examples of Levyprocesses with jumps In Section 5 we will link the outcomeswith other optimal problems such that explicit solutionsfor the new problems can also be feasible to a generalMarkov process with a large class of reward functions Finallyconcluding remarks are given in Section 6

2 Optimal Rule for Continuous-TimeMarkov Processes

For a Markov process 119883119905 the infinitesimal generator of 119883119905 isdefined as

G119892 (119909) = lim1199050

E119909119892 (119883119905) minus 119892 (119909)119905 (6)

where 119892(119909) is twice continuous differentiable In the diffusioncase namely

d119883119905 = 119886 (119883119905) d119905 + 120590 (119883119905) d119882119905 (7)

where119882119905 is a standard Brownianmotion 119886(119909) gt 0 120590(119909) = 0the infinitesimal generator is equivalent to

G119892 (119909) = 119886 (119909) d119892d119909 + 1205902 (119909)2 d2119892

d1199092 (8)

In the case of Levy process with jumps driven by the equation

d119883119905 = 119886 (119883119905minus) d119905 + 120590 (119883119905minus) d119882119905 + 119888 (119883119905minus) d119873119905 (9)

where 119873119905 is a homogeneous Poisson process 119886(119909) gt 0120590(119909) = 0 119888(119909) gt 0 the infinitesimal generator is

G119892 (119909) = 119886 (119909) d119892d119909 + 1205902 (119909)2 d2119892

d1199092+ int (119892 (119909 + 119888 (119910)) minus 119892 (119909))Π (d119910) (10)

where Π(sdot) is the Levy measure (for jump diffusion and itsgenerators eg we can refer to Gihman and Skorohod [6])

First wemake the assumption about the reward function

Assumption 1 The reward function 119891(119909) is nonincreasingconcave and twice continuous differentiable that isd119891(119909)d119909 le 0 d2119891(119909)d1199092 le 0 and 119891(119909) is 1198622

Under Assumption 1 we present the explicit solutions forgeneral continuous-time Markov processes

Theorem 2 For aMarkov process with infinitesimal generatorG let 119892(119909) be the solution of

G119892 (119909) = 119903119892 (119909) (11)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(12)

Given the reward function 119891(119909) in Assumption 1 if there existsa point 119909lowast such that

d119891 (119909)d119909

10038161003816100381610038161003816100381610038161003816119909=119909lowast = 119891 (119909lowast)119892 (119909lowast) d119892 (119909)d119909

10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (13)

then the optimal problem

V (119909) = sup120591

E119909 [eminus119903(120591minus119905)119891 (119883120591)] (14)

has an explicit expression for the value function

V (119909) = 119891 (119909) 119909 le 119909lowast119891 (119909lowast)119892 (119909lowast) 119892 (119909) 119909 ge 119909lowast (15)

The optimal stopping time is

120591lowast = inf 119904 ge 119905 119883119904 le 119909lowast (16)

Proof Define a function

ℎ (119909) = 119891 (119909lowast)119892 (119909lowast) 119892 (119909) minus 119891 (119909) (17)

Chinese Journal of Mathematics 3

and then we get

ℎ (119909) = 0dℎ (119909)d119909 = 119891 (119909lowast)119892 (119909lowast) d119892 (119909)d119909 minus d119891 (119909)

d119909 = 0iff 119909 = 119909lowast

(18)

d2ℎ (119909)d1199092 = 119891 (119909lowast)119892 (119909lowast) d

2119892 (119909)d1199092 minus d2119891 (119909)

d1199092 ge 0 (19)

Hence ℎ(119909) attains the minimum at 119909 = 119909lowast (119891(119909lowast)119892(119909lowast))119892(119909) ge 119891(119909) So we have the inequalityE119909eminus119903(120591minus119905)119891 (119883120591) le E119909eminus119903(120591minus119905)V (119883120591) (20)

The value function V(119909) is 1198621 V10158401015840(119909) exists and iscontinuous except at 119909lowast By Itorsquos formula

eminus119903119906V (119883119906) = eminus119903119905V (119883119905) + local martingale

+ int119906119905eminus119903119904 (G minus 119903) V (119883119904) d119904 (21)

for any time 119906 gt 119905 where G is the infinitesimal generator ofthe process119883119905

Because the value function V(119909) is bounded the localmartingale term on the right-hand side of (21) is alsobounded (all the other terms in (21) are clearly bounded)implying that it is in fact a martingale with zero expectationHence by the optimal sampling theorem we have for anystopping time 120591E119909eminus119903(120591minus119905)V (119883120591) = V (1198830)

+ E119909 int120591119905eminus119903(119904minus119905) (G minus 119903) V (119883119904) d119904 (22)

(a) When 119909 le 119909lowast V(119909) = 119891(119909) we claim that

(G minus 119903) V (119909) = (G minus 119903) 119891 (119909) le 0 (23)

For diffusion

(G minus 119903) 119891 (119909) = 119886 (119909) d119891d119909 + 1205902 (119909)2 d2119891

d1199092 minus 119903119891 (119909) (24)

119886(119909) gt 0 d119891(119909)d119909 le 0 d2119891(119909)d1199092 le 0 119891(119909) ge 119891(119909lowast) gt 0and the inequality in (23) holds naturally For Levy processeswith jumps

(G minus 119903) 119891 (119909) = 119886 (119909) d119891d119909 + 1205902 (119909)2 d2119891

d1199092+ int (119891 (119909 + 119888 (119910)) minus 119891 (119909))Π (d119910)minus 119903119891 (119909)

(25)

As 119888(119910) gt 0 and 119891(119909) is decreasing on states then int(119891(119909 +119888(119910)) minus 119891(119909))Π(d119910) lt 0

(b) When 119909 ge 119909lowast V(119909) = (119891(119909lowast)119892(119909lowast))119892(119909) thefunction 119892(119909) is the solution of

G119892 (119909) = 119903119892 (119909) (26)

so (G minus 119903)V(119909) = 0Therefore from (a) and (b) we have (Gminus 119903)V(119909) le 0 and

the equality holds for 119909 ge 119909lowast It turns to beE119909eminus119903(120591minus119905)119891 (119883120591) le E119909eminus119903(120591minus119905)V (119883120591)

= V (1198830)+ E119909 int120591

119905eminus119903(119904minus119905) (G minus 119903) V (119883119904) d119904

le V (119909) (27)

From which we can see that V(119909) is an upper bound for thevalue starting from1198830 = 119909 This bound is achieved when 120591 =120591lowast When the starting state 119909 is smaller than 119909lowast the optimalstopping time is 120591lowast = 119905 That is sup120591E

119909eminus119903(120591minus119905)119891(119883120591) =E119909119891(119883119905) = V(119909) as it reaches its upper bound If the startingstate is greater than the point 119909lowast it must wait until 120591lowast whichis the first hitting time to the point 119909lowast At the stopping time120591lowast 119891(119883120591lowast) = V(119883120591lowast) and int120591lowast119905 eminus119903(119904minus119905)(G minus 119903)V(119883119904)d119904 = 0 soE119909eminus119903(120591

lowastminus119905)119891(119883120591lowast) = V(119909) reaching its upper boundRemark 3 Actually for all diffusionTheorem 2 holds for thedrift term satisfying 119886(119909) lt 0 1198861015840(119909) lt 119903 We calculate that

(119886 (119909) d119891 (119909)d119909 minus 119903119891 (119909))1015840

= 1198861015840 (119909) 1198911015840 (119909) + 119886 (119909) 11989110158401015840 (119909) minus 1199031198911015840 (119909) ge 0(28)

so for 119909 le 119909lowast119886 (119909) 1198911015840 (119909) minus 119903119891 (119909) le 119886 (119909lowast) 1198911015840 (119909lowast) minus 119903119891 (119909lowast)

= 119891 (119909lowast)119892 (119909lowast) (119886 (119909lowast) 1198921015840 (119909lowast) minus 119903119892 (119909lowast)) (29)

As 119892(119909lowast) satisfies 119886(119909lowast)1198921015840(119909lowast)+ (1205902(119909lowast)2)11989210158401015840(119909lowast)minus119903119892(119909lowast) =0 we can arrive at

(G minus 119903) 119891 (119909) = 119886 (119909) 1198911015840 (119909) + 1205902 (119909)2 11989110158401015840 (119909) minus 119903119891 (119909)le minus1205902 (119909lowast)2 119891 (119909lowast)119892 (119909lowast) 11989210158401015840 (119909lowast)+ 1205902 (119909)2 11989110158401015840 (119909) le 0

(30)

The rest of the proof is the same as that in Theorem 2

However the condition for Levy process will be morecomplicated In order to have a uniform style we restrictto the case 119886(119909) gt 0 Moreover we can see from (19) thatif the point 119909lowast exists then it is unique Next we presentresults on some classical Markov processes as applications ofTheorem 2


3 Diffusion

31 Brownian Motion with Drift For Brownian motion withdrift 120583 gt 0 and variance 1205902 (120590 = 0) namely the process isdriven by the SDE

d119883119905 = 120590d119882119905 + 120583d1199051198830 = 119909 gt 0 (31)

by usingTheorem 2 we get the following proposition

Proposition 4 Let 119891(119909) be the function in Assumption 1 ifthere exists a point 119909lowast such that

d119891 (119909)d119909

10038161003816100381610038161003816100381610038161003816119909=119909lowast = minus(radic12058321205904 + 21199031205902 + 1205831205902)119891(119909lowast) 119891 (119909lowast) gt 0

(32)

then the value function V(119909) has the formV (119909)

=

119891 (119909) 119909 le 119909lowast119891 (119909lowast) exp[[(radic

12058321205904 + 21199031205902 + 1205831205902)(119909lowast minus 119909)]] 119909 ge 119909lowast(33)



Proof The infinitesimal generator for Brownian motion withdrift is G = 120583(dd119909) + (12059022)(d2d1199092) It is well known thatthe ordinary differential equation (ODE) of G119892(119909) = 119903119892(119909)has two linearly independent solutions

120601 (119909) = exp[[119909(radic12058321205904 + 21199031205902 minus 1205831205902)]]

120593 (119909) = exp[[minus119909(radic12058321205904 + 21199031205902 + 1205831205902)]]

(35)

So 119892(119909) = 1198881120601(119909) + 1198882120593(119909) (1198881 and 1198882 are constants)Considering the boundary condition lim119909rarrinfin119892(119909) = 0 and1198921015840(119909) le 0 11989210158401015840(119909) ge 0 then 1198881 must be equal to zero 119892(119909) =1198882120593(119909) and 1198882 gt 0 Equation (13) in Theorem 2 tells us thatthe point 119909lowast is determined by

1198911015840 (119909lowast) = 119891 (119909lowast)119892 (119909lowast) 1198921015840 (119909lowast)= minus(radic12058321205904 + 21199031205902 + 1205831205902)119891(119909lowast)

(36)

since 1198921015840(119909)119892(119909) = minusradic12058321205904 + 21199031205902 minus 1205831205902 Thus theexpression for the value function V(119909) is easily obtained by(15)

As the simplest example we take the reward function119891(119909) = 1 minus 119909 and we take the parameters 120583 = 1 120590 = radic2and 119903 = 1 Then the problem V(119909) = sup120591 E

119909[eminus(120591minus119905)(1 minus 119883120591)]has the solution

V (119909) =1 minus 119909 119909 le 32 minus radic52 (radic52 minus 12) exp[minus(radic52 + 12)119909 + radic52 minus 12] 119909 ge 32 minus radic52 (37)


120591lowast = inf 119904 ge 119905 119883119904 le 32 minus radic52 (38)

We draw the value function and the point 119909lowast in Figure 1

32 Geometric BrownianMotion As for geometric Brownianmotion that is the process119883119905 is driven by

d119883119905 = 119883119905 (120590d119882119905 + 120583d119905) 1198830 = 119909 gt 0 (39)

we derive the following proposition


d119891 (119909)d119909

10038161003816100381610038161003816100381610038161003816119909=119909lowast = 120573119891 (119909lowast)119909lowast 119891 (119909lowast) gt 0 (40)

where120573 = 12minus1205831205902minusradic(12 minus 1205831205902)2 + 21199031205902 then the valuefunction V(119909) has the form

V (119909) = 119891 (119909) 119909 le 119909lowast119891 (119909lowast) ( 119909119909lowast )

120573 119909 ge 119909lowast (41)




05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)

Figure 1 Plot of the value function and the point 119909lowast (119909lowast ≃ 03820)for Brownian motion with parameters 120583 = 1 120590 = radic2 and 119903 = 1

Proof The infinitesimal generator for geometric Brownianmotion is G = 120583119909(dd119909) + (120590211990922)(d2d1199092) The asso-ciated ODE of G119892(119909) = 119903119892(119909) has two linearly inde-pendent solutions 120601(119909)1205731015840 and 120593(119909)120573 where 1205731015840 = 12 minus1205831205902 + radic(12 minus 1205831205902)2 + 21199031205902 and 120573 = 12 minus 1205831205902 minusradic(12 minus 1205831205902)2 + 21199031205902 By using the boundary conditions ofTheorem 2 we find that 119892(119909) = 119888120593(119909)120573 where 119888 is a constantbigger than zeroThen by substituting 119892(119909) inTheorem 2 wededuce the value function and the point 119909lowastRemark 6 If the volatility 120590 is equal to zero then the drift 120583should be smaller than zero Or the problem cannot be solvedby Theorem 2 We take the reward function 119891(119909) = 1 minus 119909 asan illustration For 120590 = 0 and 120583 lt 0 from Theorem 2 thesolution for the optimal value function is

V (119909) = 1 minus 119909 119909 le 119903119903 minus 120583 minus120583119903 minus 120583 (119903119909 minus 120583119909119903 )119903120583 119909 ge 119903119903 minus 120583

(43)


120591lowast = inf 119905 119883119905 le 119903119903 minus 120583 (44)

If 120590 = 0 and 120583 gt 0 d119883119904 = 120583119883119904d119904 that is 119883119904 = 119909119890120583(119904minus119905) forany time 119904 ge 119905 By direct computation we find that

V (119909) = sup120591

E119909 [eminus119903(120591minus119905) (1 minus 119909119890120583(120591minus119905))]=

0 119909 ge 11 minus 119909 119909 le 1

(45)

with the optimal stopping time

120591lowast = inf 119905 119883119905 le 1 (46)

When the starting state is smaller than 1 it should take actionat once When the starting state is greater than 1 it will neverinvest because the process is increasing and it never reachesits optimal invest time But in this case the point 119909lowast = 1is not determined by (13) Because the left side is minus1 and theright side is 0 if we replace 119891(119909) = 1 minus 119909 and 119909lowast = 1 in (13)

Taking parameters 120583 = 1 120590 = radic2 and 119903 = 1 the valuefunction for the reward function 119891(119909) = 1 minus 119909 is

V (119909) = 1 minus 119909 119909 le 12 14119909 119909 ge 12

(47)


120591lowast = inf 119905 119883119905 le 12 (48)

This problemrsquos solution is the key to finding the optimum

V (119909) = sup120591

E119909 [eminus119903120591 (1198831120591 minus 1198832120591)] (49)

which has many applications in finance and it has beenconsidered for geometric Brownian motion from differentpoints of view in a variety of articles Let us just mention [1ndash4 7ndash9]

33 Ornstein-Uhlenbeck Process In this subsection we con-sider the optimal stopping problem when 119883 is an Ornstein-Uhlenbeck process

d119883119905 = 120590d119882119905 minus 120583119883119905d119905 1198830 = 119909 (50)

Proposition 7 Let 119891(119909) be as in Assumption 1 if there existsa point 119909lowast such that

d119891 (119909)d119909

10038161003816100381610038161003816100381610038161003816119909=119909lowast = minus119891 (119909lowast)119866119903120583+1 (119909lowast)119866119903120583 (119909lowast) 119891 (119909lowast) gt 0 (51)

then the value function V(119909) has the formV (119909) =

119891 (119909) 119909 le 119909lowast119891 (119909lowast) 119866119903120583 (119909)119866119903120583 (119909lowast) 119909 ge 119909lowast (52)

where

119866119903120583 (119909) = intinfin0119904119903120583minus1 exp(minus119909119904 minus 120590211990424120583 ) d119904 (53)



Proof For the Ornstein-Uhlenbeck process the infinitesimalgenerator is G = minus120583119909(dd119909) + (12059022)(d2d1199092) The ODE ofG119892(119909) = 119903119892(119909) has two linearly independent solutions

119865119903120583 (119909) = intinfin0119904119903120583minus1 exp(119909119904 minus 120590211990424120583 ) d119904

119866119903120583 (119909) = intinfin0119904119903120583minus1exp(minus119909119904 minus 120590211990424120583 ) d119904

(55)


The solution satisfying the boundary conditions inTheorem2is 119892(119909) = 119888119866119903120583(119909) where 119888 is a constant bigger than zeroThanks to the relations

119866119903120583 (119909) = 120583119909119903 119866119903120583+1 (119909) + 12059022119903119866119903120583+2 (119909) d119866119903120583 (119909)

d119909 = minus119866119903120583+1 (119909) d2119866119903120583 (119909)

d119909 = 119866119903120583+2 (119909) (56)

we obtain the value function and the point 119909lowastRegarding the reward function119891(119909) = 1minus119909 if there exists

a point 119909lowast such that 119891(119909lowast) = 1 minus 119909lowast gt 0 and 119866119903120583+1(119909lowast)(1 minus119909lowast) = 119866119903120583(119909lowast) the problem V(119909) has the formV (119909) =

1 minus 119909 119909 le 119909lowast(1 minus 119909lowast) 119866119903120583 (119909)119866119903120583 (119909lowast) 119909 ge 119909lowast (57)



When parameters are chosen as 120583 = 1 120590 = radic2 and 119903 = 1the point 119909lowast is the solution of

1198662 (119909) (1 minus 119909) = 1198661 (119909) (59)

Thanks to

1198661 (119909) = intinfin0

exp(minus119909119904 minus 11990422 ) d119904= exp(11990922 )int

infin

119909exp(minus11990422 ) d119904

1198662 (119909) = 1 minus 1199091198661 (119909) (60)

1198661(119909) and 1198662(119909) can be reduced to expressions ofgammainc(11990922 12) in Matlab By numerical calculationwe get the point 119909lowast ≃ minus01628 and we show the valuefunction in Figure 2

4 Leacutevy Processes with Jumps

41 The Jump Diffusion Jump diffusion processes are pro-cesses of the form

d119883119905 = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

119884119894) 1198830 = 119909 (61)

They are studied in Kou and Wang [10] as regards firstpassage times Here 119873119905 is a homogeneous Poisson processwith intensity rate 120582 gt 0 drift 120583 gt 0 and volatility120590 gt 0 and the jump sizes 1198841 1198842 are independentand identically distributed random variablesWe also assume

02

04

06

08

1

12

14

16

18

2

v(x)

minus05 0 05 1 15 2minus1x

Figure 2 The value function and the point 119909lowast ≃ minus01628 for theOrnstein-Uhlenbeck process with parameters 120583 = 1 120590 = radic2 and119903 = 1 with regard to the reward function 119891(119909) = 1 minus 119909

that the standard Brownian motion 119882119905 is independent of1198841 1198842 The common density of 119884 is given by

119901119884 (119910) = 120578eminus120578119910 (62)

where 120578 gt 0Proposition 8 Regarding the reward function 119891(119909) inAssumption 1 if there exists a point 119909lowast such that

d119891 (119909)d119909

10038161003816100381610038161003816100381610038161003816119909=119909lowast = 120573119891 (119909lowast) 119891 (119909lowast) gt 0 (63)

where 120573 lt 0 is the negative root of12059022 1199102 + 120583119910 + 120582 119910120578 minus 119910 minus 119903 = 0 (64)


119891 (119909) 119909 le 119909lowast119891 (119909lowast) exp (120573119909 minus 120573119909lowast) 119909 ge 119909lowast (65)



Proof In this case

(G minus 119903) V (119909) = 120583dV (119909)d119909 + 12059022 d2V (119909)

d1199092+ 120582intinfin0[V (119909 + 119910) minus V (119909)] 120578eminus120578119910d119910

minus 119903V (119909) (67)


(a) When 119909 le 119909lowast V(119909 + 119910) minus V(119909) le 0 for every 119910 gt0 d119891(119909)d119909 le 0 d2119891(119909)d1199092 le 0 for 120583 gt 0 and(G minus 119903)V(119883119904) le 0(b) When 119909 ge 119909lowast (G minus 119903)V(119909) = ((12)12059021205732 + 120583120573 +120582(120573(120578 minus 120573)) minus 119903)119891(119909lowast) exp(120573119909 minus 120573119909lowast) = 0 Since(G minus 119903)V(119909) le 0 for all 119909 the rest of the proof is the

same as inTheorem 2

Remark 9 We assume that 120583 gt 0 as Kou and Wang [10] doand we do not know whether the results hold for 120583 le 0

Let119867(119910) be defined as119867(119910) = (12)12059021199102 +120583119910+120582(119910(120578minus119910)) and it has three roots 1199101 = 0 1199102 = 1205782 minus 1205831205902 +radic(1205782 + 1205831205902)2 + 21205821205902 gt 0 and 1199103 = 1205782 minus 1205831205902 minusradic(1205782 + 1205831205902)2 + 21205821205902 lt 0

Next we prove that the equation 119867(119910) = 119903 for all 119903 gt 0has exactly a negative root 120573

1198671015840 (119910) = 1205902119910 + 120583 + 120582120578(120578 minus 119910)2 11986710158401015840 (119910) = 1205902 + 2120582120578(120578 minus 119910)3

(68)

so 119867(119910) is increased and convex on the interval (0 120578) with119867(0) = 0 and 119867(120578minus) = infin and there is exactly one root for119867(119910) = 119903 Furthermore since 119867(120578+) = minusinfin 119867(infin) = infinthere is at least one root on (120578infin) Similarly there is at leastone root on (minusinfin 0) as 119867(minusinfin) = infin and 119867(0) = 0 Butthe equation119867(119910) = 119903 is actually a polynomial equation withdegree three therefore it can have at most three real rootsIt follows that on each interval (minusinfin 0) and (120578infin) there isexactly one root

As an example we take the reward function 119891(119909) = 1 minus119909 again From Proposition 8 119909lowast = 1 + 1120573 where 120573 is thenegative root of

1212059021199102 + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (69)

and the value function V(119909) has the formV (119909) =

1 minus 119909 119909 le 119909lowast(1 minus 119909lowast) exp (120573119909 minus 120573119909lowast) 119909 ge 119909lowast (70)



If the parameters are 120583 = 1 120590 = radic2 119903 = 1 120582 = 1 and 120578 =15 we show the value function and the point 119909lowast in Figure 3Numerical calculation gets the negative solution for 120573 to be120573 ≃ minus18421 and the point 119909lowast ≃ 04571 Compared withthe case of Brownian motion with drift the decision point 119909lowastresults larger after adding the jump process Intuitively as risk

0

01

02

03

04

05

06

07

08

09

v(x)

05 1 15 20x

Figure 3The value function and the point 119909lowast (119909lowast ≃ 04571) for thejump diffusion model with parameters 120583 = 1 120590 = radic2 119903 = 1 120582 = 1and 120578 = 15in the system is increasing the investors are intent to wait andobserve for a longer time so that the optimal stopping occurslater and the decision point gets greater

The more general problem when the common density of119884 is given by

119901119884 (119910) = 119898sum119894=1

119901119894120578119894eminus120578119894119910 (72)

where 0 lt 1205781 lt 1205782 lt sdot sdot sdot lt 120578119898 lt infin and sum119898119894=1 119901119894 = 1 has thefollowing result


d119891 (119909)d119909


where 120573 lt 0 is the negative root of12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 minus 119903 = 0 (74)





Remark 11 The equation (12059022)1199102 + 120583119910 + 120582sum119898119894=1 119901119894(119910(120578119894 minus119910)) minus 119903 = 0 has (119898 + 2) roots which are all real and distinctIndeed it has119898+1 positive roots1205731 120573119898+1 and a negativeroot 120573 as follows

minusinfin lt 120573 lt 0 lt 1205731 lt sdot sdot sdot lt 120573119898+1 lt infin (77)


Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)

and we have 119871(0) = 0 119871(1205781minus) = infin 119871(1205781+) = minusinfin 119871(1205782minus) =infin 119871(120578119898+) = minusinfin 119871(infin) = infin and 119871(119910) is contin-uous and increasing in every interval (0 1205781minus) (1205781+1205782minus) (120578119898+infin) So it has 119898 + 1 positive roots if it alsohas a negative root 120573 then the negative root is unique as ithas at most119898+ 2 real roots The detailed proof can be foundinTheorem 31 of Kou and Cai [11]

42The Exponential Levy-Type Stochastic Integral The expo-nential Levy-type stochastic integral119883119905 is given by

d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)

Here 119873119905 is a Poisson process with intensity rate 120582 gt 0 and120583 and 120590 are positive constants The jump sizes 1198841 1198842 are independent and identically distributed random variablesand independent of119882119905 The common density of 119884 is given by

119901119884 (119910) = 120578eminus120578119910 (80)

where 120578 gt 1 ensures that119883 has a finite expectation

Proposition 12 Given the reward function 119891(119909) in Assump-tion 1 if there exists a point 119909lowast such that

d119891 (119909)d119909


where 120573 lt 0 is the negative root of12059022 1199102 (119910 minus 1) + 120583119910 + 120582 119910120578 minus 119910 minus 119903 = 0 (82)


119891 (119909) 119909 le 119909lowast119891 (119909lowast) ( 119909119909lowast )

120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)

d1199092+ 120582intinfin0[V (119909e119910) minus V (119909)] 120578eminus120578119910d119910

minus 119903V (119909) (85)

(a) When 119909 le 119909lowast because 119910 ge 0 and e119910 ge 1 then119909e119910 ge 119909 and 119891(119909e119910) le 119891(119909) Moreover thanks to120583 gt 0 d119891(119909)d119909 le 0 d2119891(119909)d1199092 le 0 we have(G minus 119903)V(119883119904) le 0(b) When 119909 ge 119909lowast

(G minus 119903) V (119909)= [121205902 (1205732 minus 120573) + 120583120573 + 120582 120573120578 minus 120573 minus 119903]119891 (119909lowast) ( 119909119909lowast )

120573

= 0(86)

Taking (a) and (b) together we conclude that (G minus119903)V(119909) le 0 for every 119909 gt 0 Hence V(119909) is the upperbound for E119909eminus119903(120591minus119905)119891(119883120591) At the optimal stopping time 120591lowastE119909eminus119903(120591

lowastminus119905)119891(119883120591lowast) = V(119909) attaining the upper boundRemark 13 The equation (12059022)(119910 minus 1) + 120583119910 + 120582(119910(120578 minus 119910)) minus119903 = 0 has a unique negative root 120573 This can be obtainedproceeding as in Remark 11

For the reward function 119891(119909) = 1 minus 119909 120573 is the negativeroot of

12059022 119910 (119910 minus 1) + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (87)

we find 119909lowast = 120573(120573 minus 1) and the value function is

V (119909) = 1 minus 119909 119909 le 119909lowast(1 minus 119909lowast) ( 119909119909lowast )

120573 119909 ge 119909lowast (88)



For the choice of parameters 120583 = 1 120590 = radic2 119903 = 1 120582 = 1 and120578 = 15 we obtain 120573 ≃ minus12020 and the point 119909lowast ≃ 05459and we draw the graph of the value function in Figure 4The point 119909lowast is bigger than the value concerning geometricBrownian motion after adding the jump risk

5 Connection with the OtherOptimal Problems

As the reward function 119891(119909) is 1198622 in Assumption 1 by Itorsquosformula

eminus119903(119906minus119905)119891 (119883119905) = 119891 (119909)+ int119906119905eminus119903(119904minus119905) [G119891 (119883119904) minus 119903119891 (119883119904)] d119904

+ local martingale(90)


05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)

Figure 4 The value function and the point 119909lowast (119909lowast ≃ 05459) for120583 = 1 120590 = radic2 119903 = 1 120582 = 1 and 120578 = 15

Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)

where 120574 = 0 is a constant Taking expectation in (90)

E119909eminus119903(119906minus119905)119891 (119883119906) = 119891 (119909) + E119909 int119906119905eminus119903(119904minus119905)120574119891 (119883119904) d119904

+ E119909 (local martingale) (92)

As assumed in the paper eminus119903(119906minus119905)119891(119883119905) is bounded soE119909(local martingale) = 0 For a stopping time 120591

E119909eminus119903(120591minus119905)119891 (119883120591) = 119891 (119909) + E119909 int120591119905eminus119903(119904minus119905)120574119891 (119883119904) d119904 (93)

from which we can see that if 120574 gt 0 the problemsup120591 E

119909eminus119903(120591minus119905)119891(119883120591) is equivalent tosup120591

E119909 int120591119905eminus119903(119904minus119905)119891 (119883119904) d119904 (94)

and if 120574 lt 0 the problem sup120591 E119909eminus119903(120591minus119905)119891(119883120591) is equivalent

to

inf120591E119909 int120591119905eminus119903(119904minus119905)119891 (119883119904) d119904 (95)

We conclude the solution for the problemssup120591 E

119909 int120591119905eminus119903(119904minus119905)119891(119883119904)d119904 and inf120591 E

119909 int120591119905eminus119903(119904minus119905)119891(119883119904)d119904

in the following theorem

Theorem 14 For a given reward function 119891(119909) in Assump-tion 1 ifG119891(119909) minus 119903119891(119909) = 120574119891(119909) and 120574 gt 0 the solution for

119908 (119909) = sup120591


has the form

119908 (119909) = 0 119909 le 119909lowast119891 (119909lowast)119892 (119909lowast) 119892 (119909)120574 minus 119891 (119909)120574 119909 ge 119909lowast (97)



If 120574 lt 0 then the solution for

119908 (119909) = inf120591E119909 int120591119905eminus119903(119904minus119905)119891 (119883119904) d119904 (99)

has the form




The function 119892(119909) is given by

G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)

and the point 119909lowast is determined by

d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)

Since the expression of E119909 int120591119905eminus119903(119904minus119905)119891(119883119904)d119904 can be rep-

resented as valuation of equity (see A3 in Duffie [12]) wewill consider the practical application in the stock market ina future work

6 Conclusions

In this paper we have studied the optimal stopping problemsfor a continuous-timeMarkov process andwehave presentedthe explicit value function and optimal rule

Three main contributions of this paper are as followsFirst we constructed the value functions and optimal rulesfor the problems with a large class of reward functions whichare different from the previous researches using some specificfunctions With simple conditions for the reward functionswe gave a rigorous mathematical proof to deduce the optimal


rule based on variational inequalities Second we derivedthe solution of the problem for a general Markov processMeanwhile we showed the explicit value functions andoptimal stopping times for some concrete Markov processesincluding diffusion and Levy processes with jumps Finallywe linked our results to some optimal problems Also weextended the explicit solution of the new optimal problem toa broad class of reward functions and to a general continuous-time Markov process

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] S T Wong ldquoThe generalized perpetual American exchange-option problemrdquo Advances in Applied Probability vol 40 no1 pp 163ndash182 2008

[2] R McDonald and D Siegel ldquoThe value of waiting to investrdquoQuarterly Journal of Economics vol 101 no 4 pp 707ndash727 1986

[3] Y Hu and B Oslashksendal ldquoOptimal time to invest when theprice processes are geometric Brownian motionsrdquo Finance andStochastics vol 2 no 3 pp 295ndash310 1998

[4] K Nishide and L C Rogers ldquoOptimal time to exchange twobasketsrdquo Journal of Applied Probability vol 48 no 1 pp 21ndash302011

[5] J L Pedersen and G Peskir ldquoSolving non-linear optimalstopping problems by the method of time-changerdquo StochasticAnalysis and Applications vol 18 no 5 pp 811ndash835 2000

[6] I I Gihman and A V Skorohod Stochastic Differential Equa-tions The Theory of Stochastic Processes III Springer-VerlagNew York NY USA 1979

[7] S Christensen and A Irle ldquoA harmonic function technique forthe optimal stopping of diffusionsrdquo Stochastics vol 83 no 4-6pp 347ndash363 2011

[8] K Helmes and R H Stockbridge ldquoConstruction of thevalue function and optimal rules in optimal stopping of one-dimensional diffusionsrdquo Advances in Applied Probability vol42 no 1 pp 158ndash182 2010

[9] T E Olsen and G Stensland ldquoOn optimal timing of investmentwhen cost components are additive and follow geometricdiffusionsrdquo Journal of Economic Dynamics and Control vol 16no 1 pp 39ndash51 1992

[10] S G Kou andHWang ldquoFirst passage times of a jump diffusionprocessrdquoAdvances in Applied Probability vol 35 no 2 pp 504ndash531 2003

[11] N Cai and S G Kou ldquoOption pricing under a mixed-exponential jump diffusion modelrdquo Management Science vol57 no 11 pp 2067ndash2081 2011

[12] D Duffie ldquoCredit risk modeling with affine processesrdquo Journalof Banking and Finance vol 29 no 11 pp 2751ndash2802 2005

Submit your manuscripts athttpswwwhindawicom

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MathematicsJournal of


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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Stochastic AnalysisInternational Journal of


powerful in the following proof as we will see Secondlywe pay attention to general Markov processes includingdiffusion and Levy processes with jumpsWith the help of theinfinitesimal generator we obtain an explicit formula for thevalue function and the stopping timeThirdly we find that theoptimal problem (2) is equivalent to other optimal problemslike

119908 (119909) = sup120591

E119909 [int120591119905eminus119903(119904minus119905)119891 (119883119904) d119904] (4)

119908 (119909) = inf120591E119909 [int120591119905eminus119903(119904minus119905)119891 (119883119904) d119904] (5)

This work is inspired by Pedersen and Peskir [5] who verifiedthe equivalence of problems (2) and (4) for 119891(119909) = 119909 where119883119905 is an Ornstein-Uhlenbeck process Note that we do notconsider the case 119891(119909) = 119909 so our approach to deal with theoptimal problem is different from that of Pedersen and PeskirMoreover our work naturally explore the explicit solutionsof the new optimal problem (4) for a larger class of rewardfunctions and underlying processes119883119905

The paper is organized as follows In Section 2 theexplicit value function and optimal stopping time are derivedfor a general Markov process along with the condition forthe reward functions Section 3 discusses some applicationsto diffusion which include Brownian motion with driftgeometric Brownian motion and the Ornstein-Uhlenbeckprocess Section 4 displays some concrete examples of Levyprocesses with jumps In Section 5 we will link the outcomeswith other optimal problems such that explicit solutionsfor the new problems can also be feasible to a generalMarkov process with a large class of reward functions Finallyconcluding remarks are given in Section 6

2 Optimal Rule for Continuous-TimeMarkov Processes

For a Markov process 119883119905 the infinitesimal generator of 119883119905 isdefined as

G119892 (119909) = lim1199050

E119909119892 (119883119905) minus 119892 (119909)119905 (6)

where 119892(119909) is twice continuous differentiable In the diffusioncase namely

d119883119905 = 119886 (119883119905) d119905 + 120590 (119883119905) d119882119905 (7)

where119882119905 is a standard Brownianmotion 119886(119909) gt 0 120590(119909) = 0the infinitesimal generator is equivalent to

G119892 (119909) = 119886 (119909) d119892d119909 + 1205902 (119909)2 d2119892

d1199092 (8)

In the case of Levy process with jumps driven by the equation

d119883119905 = 119886 (119883119905minus) d119905 + 120590 (119883119905minus) d119882119905 + 119888 (119883119905minus) d119873119905 (9)

where 119873119905 is a homogeneous Poisson process 119886(119909) gt 0120590(119909) = 0 119888(119909) gt 0 the infinitesimal generator is

G119892 (119909) = 119886 (119909) d119892d119909 + 1205902 (119909)2 d2119892

d1199092+ int (119892 (119909 + 119888 (119910)) minus 119892 (119909))Π (d119910) (10)

where Π(sdot) is the Levy measure (for jump diffusion and itsgenerators eg we can refer to Gihman and Skorohod [6])

First wemake the assumption about the reward function

Assumption 1 The reward function 119891(119909) is nonincreasingconcave and twice continuous differentiable that isd119891(119909)d119909 le 0 d2119891(119909)d1199092 le 0 and 119891(119909) is 1198622

Under Assumption 1 we present the explicit solutions forgeneral continuous-time Markov processes

Theorem 2 For aMarkov process with infinitesimal generatorG let 119892(119909) be the solution of

G119892 (119909) = 119903119892 (119909) (11)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(12)

Given the reward function 119891(119909) in Assumption 1 if there existsa point 119909lowast such that

d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (13)

then the optimal problem

V (119909) = sup120591

E119909 [eminus119903(120591minus119905)119891 (119883120591)] (14)

has an explicit expression for the value function

V (119909) = 119891 (119909) 119909 le 119909lowast119891 (119909lowast)119892 (119909lowast) 119892 (119909) 119909 ge 119909lowast (15)



Proof Define a function

ℎ (119909) = 119891 (119909lowast)119892 (119909lowast) 119892 (119909) minus 119891 (119909) (17)


and then we get


d119909 = 0iff 119909 = 119909lowast

(18)


2119892 (119909)d1199092 minus d2119891 (119909)

d1199092 ge 0 (19)










For diffusion

(G minus 119903) 119891 (119909) = 119886 (119909) d119891d119909 + 1205902 (119909)2 d2119891

d1199092 minus 119903119891 (119909) (24)


(G minus 119903) 119891 (119909) = 119886 (119909) d119891d119909 + 1205902 (119909)2 d2119891


(25)



G119892 (119909) = 119903119892 (119909) (26)



= V (1198830)+ E119909 int120591


le V (119909) (27)




(119886 (119909) d119891 (119909)d119909 minus 119903119891 (119909))1015840

= 1198861015840 (119909) 1198911015840 (119909) + 119886 (119909) 11989110158401015840 (119909) minus 1199031198911015840 (119909) ge 0(28)





(30)




3 Diffusion


d119883119905 = 120590d119882119905 + 120583d1199051198830 = 119909 gt 0 (31)



d119891 (119909)d119909


(32)


=






120601 (119909) = exp[[119909(radic12058321205904 + 21199031205902 minus 1205831205902)]]

120593 (119909) = exp[[minus119909(radic12058321205904 + 21199031205902 + 1205831205902)]]

(35)



(36)









d119883119905 = 119883119905 (120590d119882119905 + 120583d119905) 1198830 = 119909 gt 0 (39)



d119891 (119909)d119909




120573 119909 ge 119909lowast (41)




05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)




(43)




V (119909) = sup120591


0 119909 ge 11 minus 119909 119909 le 1

(45)


120591lowast = inf 119905 119883119905 le 1 (46)



V (119909) = 1 minus 119909 119909 le 12 14119909 119909 ge 12

(47)


120591lowast = inf 119905 119883119905 le 12 (48)


V (119909) = sup120591

E119909 [eminus119903120591 (1198831120591 minus 1198832120591)] (49)



d119883119905 = 120590d119882119905 minus 120583119883119905d119905 1198830 = 119909 (50)


d119891 (119909)d119909




where







(55)



119866119903120583 (119909) = 120583119909119903 119866119903120583+1 (119909) + 12059022119903119866119903120583+2 (119909) d119866119903120583 (119909)

d119909 = minus119866119903120583+1 (119909) d2119866119903120583 (119909)

d119909 = 119866119903120583+2 (119909) (56)







1198662 (119909) (1 minus 119909) = 1198661 (119909) (59)

Thanks to

1198661 (119909) = intinfin0


infin

119909exp(minus11990422 ) d119904

1198662 (119909) = 1 minus 1199091198661 (119909) (60)




d119883119905 = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

119884119894) 1198830 = 119909 (61)


02

04

06

08

1

12

14

16

18

2

v(x)




119901119884 (119910) = 120578eminus120578119910 (62)


d119891 (119909)d119909







Proof In this case

(G minus 119903) V (119909) = 120583dV (119909)d119909 + 12059022 d2V (119909)


minus 119903V (119909) (67)



same as inTheorem 2




1198671015840 (119910) = 1205902119910 + 120583 + 120582120578(120578 minus 119910)2 11986710158401015840 (119910) = 1205902 + 2120582120578(120578 minus 119910)3

(68)



1212059021199102 + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (69)






0

01

02

03

04

05

06

07

08

09

v(x)

05 1 15 20x



119901119884 (119910) = 119898sum119894=1

119901119894120578119894eminus120578119894119910 (72)



d119891 (119909)d119909



119898sum119894=1

119901119894 119910120578119894 minus 119910 minus 119903 = 0 (74)








Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)



d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)


119901119884 (119910) = 120578eminus120578119910 (80)



d119891 (119909)d119909





120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)


minus 119903V (119909) (85)



120573

= 0(86)







120573 119909 ge 119909lowast (88)









05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





and then we get


d119909 = 0iff 119909 = 119909lowast

(18)


2119892 (119909)d1199092 minus d2119891 (119909)

d1199092 ge 0 (19)










For diffusion

(G minus 119903) 119891 (119909) = 119886 (119909) d119891d119909 + 1205902 (119909)2 d2119891

d1199092 minus 119903119891 (119909) (24)


(G minus 119903) 119891 (119909) = 119886 (119909) d119891d119909 + 1205902 (119909)2 d2119891


(25)



G119892 (119909) = 119903119892 (119909) (26)



= V (1198830)+ E119909 int120591


le V (119909) (27)




(119886 (119909) d119891 (119909)d119909 minus 119903119891 (119909))1015840

= 1198861015840 (119909) 1198911015840 (119909) + 119886 (119909) 11989110158401015840 (119909) minus 1199031198911015840 (119909) ge 0(28)





(30)




3 Diffusion


d119883119905 = 120590d119882119905 + 120583d1199051198830 = 119909 gt 0 (31)



d119891 (119909)d119909


(32)


=






120601 (119909) = exp[[119909(radic12058321205904 + 21199031205902 minus 1205831205902)]]

120593 (119909) = exp[[minus119909(radic12058321205904 + 21199031205902 + 1205831205902)]]

(35)



(36)









d119883119905 = 119883119905 (120590d119882119905 + 120583d119905) 1198830 = 119909 gt 0 (39)



d119891 (119909)d119909




120573 119909 ge 119909lowast (41)




05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)




(43)




V (119909) = sup120591


0 119909 ge 11 minus 119909 119909 le 1

(45)


120591lowast = inf 119905 119883119905 le 1 (46)



V (119909) = 1 minus 119909 119909 le 12 14119909 119909 ge 12

(47)


120591lowast = inf 119905 119883119905 le 12 (48)


V (119909) = sup120591

E119909 [eminus119903120591 (1198831120591 minus 1198832120591)] (49)



d119883119905 = 120590d119882119905 minus 120583119883119905d119905 1198830 = 119909 (50)


d119891 (119909)d119909




where







(55)



119866119903120583 (119909) = 120583119909119903 119866119903120583+1 (119909) + 12059022119903119866119903120583+2 (119909) d119866119903120583 (119909)

d119909 = minus119866119903120583+1 (119909) d2119866119903120583 (119909)

d119909 = 119866119903120583+2 (119909) (56)







1198662 (119909) (1 minus 119909) = 1198661 (119909) (59)

Thanks to

1198661 (119909) = intinfin0


infin

119909exp(minus11990422 ) d119904

1198662 (119909) = 1 minus 1199091198661 (119909) (60)




d119883119905 = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

119884119894) 1198830 = 119909 (61)


02

04

06

08

1

12

14

16

18

2

v(x)




119901119884 (119910) = 120578eminus120578119910 (62)


d119891 (119909)d119909







Proof In this case

(G minus 119903) V (119909) = 120583dV (119909)d119909 + 12059022 d2V (119909)


minus 119903V (119909) (67)



same as inTheorem 2




1198671015840 (119910) = 1205902119910 + 120583 + 120582120578(120578 minus 119910)2 11986710158401015840 (119910) = 1205902 + 2120582120578(120578 minus 119910)3

(68)



1212059021199102 + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (69)






0

01

02

03

04

05

06

07

08

09

v(x)

05 1 15 20x



119901119884 (119910) = 119898sum119894=1

119901119894120578119894eminus120578119894119910 (72)



d119891 (119909)d119909



119898sum119894=1

119901119894 119910120578119894 minus 119910 minus 119903 = 0 (74)








Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)



d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)


119901119884 (119910) = 120578eminus120578119910 (80)



d119891 (119909)d119909





120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)


minus 119903V (119909) (85)



120573

= 0(86)







120573 119909 ge 119909lowast (88)









05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





3 Diffusion


d119883119905 = 120590d119882119905 + 120583d1199051198830 = 119909 gt 0 (31)



d119891 (119909)d119909


(32)


=






120601 (119909) = exp[[119909(radic12058321205904 + 21199031205902 minus 1205831205902)]]

120593 (119909) = exp[[minus119909(radic12058321205904 + 21199031205902 + 1205831205902)]]

(35)



(36)









d119883119905 = 119883119905 (120590d119882119905 + 120583d119905) 1198830 = 119909 gt 0 (39)



d119891 (119909)d119909




120573 119909 ge 119909lowast (41)




05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)




(43)




V (119909) = sup120591


0 119909 ge 11 minus 119909 119909 le 1

(45)


120591lowast = inf 119905 119883119905 le 1 (46)



V (119909) = 1 minus 119909 119909 le 12 14119909 119909 ge 12

(47)


120591lowast = inf 119905 119883119905 le 12 (48)


V (119909) = sup120591

E119909 [eminus119903120591 (1198831120591 minus 1198832120591)] (49)



d119883119905 = 120590d119882119905 minus 120583119883119905d119905 1198830 = 119909 (50)


d119891 (119909)d119909




where







(55)



119866119903120583 (119909) = 120583119909119903 119866119903120583+1 (119909) + 12059022119903119866119903120583+2 (119909) d119866119903120583 (119909)

d119909 = minus119866119903120583+1 (119909) d2119866119903120583 (119909)

d119909 = 119866119903120583+2 (119909) (56)







1198662 (119909) (1 minus 119909) = 1198661 (119909) (59)

Thanks to

1198661 (119909) = intinfin0


infin

119909exp(minus11990422 ) d119904

1198662 (119909) = 1 minus 1199091198661 (119909) (60)




d119883119905 = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

119884119894) 1198830 = 119909 (61)


02

04

06

08

1

12

14

16

18

2

v(x)




119901119884 (119910) = 120578eminus120578119910 (62)


d119891 (119909)d119909







Proof In this case

(G minus 119903) V (119909) = 120583dV (119909)d119909 + 12059022 d2V (119909)


minus 119903V (119909) (67)



same as inTheorem 2




1198671015840 (119910) = 1205902119910 + 120583 + 120582120578(120578 minus 119910)2 11986710158401015840 (119910) = 1205902 + 2120582120578(120578 minus 119910)3

(68)



1212059021199102 + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (69)






0

01

02

03

04

05

06

07

08

09

v(x)

05 1 15 20x



119901119884 (119910) = 119898sum119894=1

119901119894120578119894eminus120578119894119910 (72)



d119891 (119909)d119909



119898sum119894=1

119901119894 119910120578119894 minus 119910 minus 119903 = 0 (74)








Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)



d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)


119901119884 (119910) = 120578eminus120578119910 (80)



d119891 (119909)d119909





120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)


minus 119903V (119909) (85)



120573

= 0(86)







120573 119909 ge 119909lowast (88)









05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)




(43)




V (119909) = sup120591


0 119909 ge 11 minus 119909 119909 le 1

(45)


120591lowast = inf 119905 119883119905 le 1 (46)



V (119909) = 1 minus 119909 119909 le 12 14119909 119909 ge 12

(47)


120591lowast = inf 119905 119883119905 le 12 (48)


V (119909) = sup120591

E119909 [eminus119903120591 (1198831120591 minus 1198832120591)] (49)



d119883119905 = 120590d119882119905 minus 120583119883119905d119905 1198830 = 119909 (50)


d119891 (119909)d119909




where







(55)



119866119903120583 (119909) = 120583119909119903 119866119903120583+1 (119909) + 12059022119903119866119903120583+2 (119909) d119866119903120583 (119909)

d119909 = minus119866119903120583+1 (119909) d2119866119903120583 (119909)

d119909 = 119866119903120583+2 (119909) (56)







1198662 (119909) (1 minus 119909) = 1198661 (119909) (59)

Thanks to

1198661 (119909) = intinfin0


infin

119909exp(minus11990422 ) d119904

1198662 (119909) = 1 minus 1199091198661 (119909) (60)




d119883119905 = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

119884119894) 1198830 = 119909 (61)


02

04

06

08

1

12

14

16

18

2

v(x)




119901119884 (119910) = 120578eminus120578119910 (62)


d119891 (119909)d119909







Proof In this case

(G minus 119903) V (119909) = 120583dV (119909)d119909 + 12059022 d2V (119909)


minus 119903V (119909) (67)



same as inTheorem 2




1198671015840 (119910) = 1205902119910 + 120583 + 120582120578(120578 minus 119910)2 11986710158401015840 (119910) = 1205902 + 2120582120578(120578 minus 119910)3

(68)



1212059021199102 + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (69)






0

01

02

03

04

05

06

07

08

09

v(x)

05 1 15 20x



119901119884 (119910) = 119898sum119894=1

119901119894120578119894eminus120578119894119910 (72)



d119891 (119909)d119909



119898sum119894=1

119901119894 119910120578119894 minus 119910 minus 119903 = 0 (74)








Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)



d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)


119901119884 (119910) = 120578eminus120578119910 (80)



d119891 (119909)d119909





120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)


minus 119903V (119909) (85)



120573

= 0(86)







120573 119909 ge 119909lowast (88)









05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119866119903120583 (119909) = 120583119909119903 119866119903120583+1 (119909) + 12059022119903119866119903120583+2 (119909) d119866119903120583 (119909)

d119909 = minus119866119903120583+1 (119909) d2119866119903120583 (119909)

d119909 = 119866119903120583+2 (119909) (56)







1198662 (119909) (1 minus 119909) = 1198661 (119909) (59)

Thanks to

1198661 (119909) = intinfin0


infin

119909exp(minus11990422 ) d119904

1198662 (119909) = 1 minus 1199091198661 (119909) (60)




d119883119905 = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

119884119894) 1198830 = 119909 (61)


02

04

06

08

1

12

14

16

18

2

v(x)




119901119884 (119910) = 120578eminus120578119910 (62)


d119891 (119909)d119909







Proof In this case

(G minus 119903) V (119909) = 120583dV (119909)d119909 + 12059022 d2V (119909)


minus 119903V (119909) (67)



same as inTheorem 2




1198671015840 (119910) = 1205902119910 + 120583 + 120582120578(120578 minus 119910)2 11986710158401015840 (119910) = 1205902 + 2120582120578(120578 minus 119910)3

(68)



1212059021199102 + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (69)






0

01

02

03

04

05

06

07

08

09

v(x)

05 1 15 20x



119901119884 (119910) = 119898sum119894=1

119901119894120578119894eminus120578119894119910 (72)



d119891 (119909)d119909



119898sum119894=1

119901119894 119910120578119894 minus 119910 minus 119903 = 0 (74)








Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)



d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)


119901119884 (119910) = 120578eminus120578119910 (80)



d119891 (119909)d119909





120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)


minus 119903V (119909) (85)



120573

= 0(86)







120573 119909 ge 119909lowast (88)









05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






same as inTheorem 2




1198671015840 (119910) = 1205902119910 + 120583 + 120582120578(120578 minus 119910)2 11986710158401015840 (119910) = 1205902 + 2120582120578(120578 minus 119910)3

(68)



1212059021199102 + 120583119910 + 120582119910120578 minus 119910 minus 119903 = 0 (69)






0

01

02

03

04

05

06

07

08

09

v(x)

05 1 15 20x



119901119884 (119910) = 119898sum119894=1

119901119894120578119894eminus120578119894119910 (72)



d119891 (119909)d119909



119898sum119894=1

119901119894 119910120578119894 minus 119910 minus 119903 = 0 (74)








Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)



d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)


119901119884 (119910) = 120578eminus120578119910 (80)



d119891 (119909)d119909





120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)


minus 119903V (119909) (85)



120573

= 0(86)







120573 119909 ge 119909lowast (88)









05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Define 119871(119910) as119871 (119910) ≜ 12059022 1199102 + 120583119910 + 120582

119898sum119894=1

119901119894 119910120578119894 minus 119910 (78)



d119883119905119883119905minus = 120590d119882119905 + 120583d119905 + d(119873119905sum119894=1

(e119884119894 minus 1)) 1198830 = 119909 gt 0

(79)


119901119884 (119910) = 120578eminus120578119910 (80)



d119891 (119909)d119909





120573 119909 ge 119909lowast (83)



Proof In this case

(G minus 119903) V (119909) = 120583119909dV (119909)d119909 + 120590211990922 d2V (119909)


minus 119903V (119909) (85)



120573

= 0(86)







120573 119909 ge 119909lowast (88)









05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





05 1 15 20x

0

01

02

03

04

05

06

07

08

09

v(x)


Assume that

G119891 (119883119904) minus 119903119891 (119883119904) = 120574119891 (119883119904) (91)










to







119908 (119909) = sup120591


has the form






has the form





G119892 (119909) = 119903119892 (119909) (102)

satisfying

lim119909rarrinfin

119892 (119909) = 0d119892 (119909)d119909 le 0

d2119892 (119909)d1199092 ge 0

(103)


d119891 (119909)d119909


10038161003816100381610038161003816100381610038161003816119909=119909lowast 119891 (119909lowast) gt 0 (104)



6 Conclusions







References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




value function and optimal rule on the optimal stopping...

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