Research Article An Optimal Stopping Problem for Jump...
6
Research Article An Optimal Stopping Problem for Jump Diffusion Logistic Population Model Yang Sun 1 and Xiaohui Ai 2 1 School of Applied Science, Harbin University of Science and Technology, Harbin 150080, China 2 Department of Mathematics, Northeast Forestry University, Harbin 150040, China Correspondence should be addressed to Yang Sun; [email protected] Received 11 May 2016; Revised 13 July 2016; Accepted 21 July 2016 Academic Editor: Khaled Bahlali Copyright © 2016 Y. Sun and X. Ai. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper examines an optimal stopping problem for the stochastic (Wiener-Poisson) jump diffusion logistic population model. We present an explicit solution to an optimal stopping problem of the stochastic (Wiener-Poisson) jump diffusion logistic population model by applying the smooth pasting technique (Dayanik and Karatzas, 2003; Dixit, 1993). We formulate this as an optimal stopping problem of maximizing the expected reward. We express the critical state of the optimal stopping region and the optimal value function explicitly. 1. Introduction e theory of optimal stopping is widely applied in many fields such as finance, insurance, and bioeconomics. Optimal stopping problems for lots of models have been put forward to meet the actual need. Bioeconomic resource models incor- porating random fluctuations in either population size or model parameters have been the subject of much interest. e optimal stopping problem is very important in mathematical bioeconomics and has been extensively studied;see Clark [1], Dayanik and Karatzas [2], Dai and Kwok [3], Presman and Sonin [4], Christensen and Irle [5], and so forth. A very classic and successful model for population growth in mathematics is logistic model = ( − 2 ) , (1) where denotes the density of resource population at time , >0 is called the intrinsic growth rate, and = / > 0 ( is the environmental carrying capacity). e logistic model is used widely to real data; however, it is too simple to provide a better simulation of the real world since there are some uncertainties, such as environment and financial effect, modeled by Gaussian white noise. Hence, the stochastic logistic differential equation is introduced to handle these problems; that is, = ( − 2 ) + , 0 = ≥ 0, ≥ 0, (2) where the constants , are mentioned in (1), is a measure of the size of the noise in the system, and is 1-dimensional Brownian motion defined on a complete probability space (Ω, F,{F } ≥0 , P) satisfing the usual conditions. ere are so many extensive researches in literature, such as Lungu and Øksendal [6], Sun and Wang [7], Liu and Wang [8], and Liu and Wang [9, 10]. Furthermore, large and sudden fluctuations in environ- mental fluctuations can not modeled by the Gaussian white noise, for examples, hurricanes, disasters, and crashes. A Poisson jump stochastic equation can explain the sudden changes. In this paper, we will concentrate on the stochastic logistic population model with Poisson jump = ( − 2 ) + +∫ ∞ 0 − (, ) , 0 = ≥ 0, ≥ 0, (3) Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 5839672, 5 pages http://dx.doi.org/10.1155/2016/5839672
Transcript of Research Article An Optimal Stopping Problem for Jump...
Research ArticleAn Optimal Stopping Problem for Jump Diffusion LogisticPopulation Model
Yang Sun1 and Xiaohui Ai2
1School of Applied Science Harbin University of Science and Technology Harbin 150080 China2Department of Mathematics Northeast Forestry University Harbin 150040 China
Correspondence should be addressed to Yang Sun sunysy126com
Received 11 May 2016 Revised 13 July 2016 Accepted 21 July 2016
Academic Editor Khaled Bahlali
Copyright copy 2016 Y Sun and X Ai This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper examines an optimal stopping problem for the stochastic (Wiener-Poisson) jumpdiffusion logistic populationmodelWepresent an explicit solution to an optimal stopping problem of the stochastic (Wiener-Poisson) jump diffusion logistic populationmodel by applying the smooth pasting technique (Dayanik and Karatzas 2003 Dixit 1993) We formulate this as an optimalstopping problem of maximizing the expected reward We express the critical state of the optimal stopping region and the optimalvalue function explicitly
1 Introduction
The theory of optimal stopping is widely applied in manyfields such as finance insurance and bioeconomics Optimalstopping problems for lots of models have been put forwardtomeet the actual need Bioeconomic resourcemodels incor-porating random fluctuations in either population size ormodel parameters have been the subject ofmuch interestTheoptimal stopping problem is very important in mathematicalbioeconomics and has been extensively studiedsee Clark [1]Dayanik and Karatzas [2] Dai and Kwok [3] Presman andSonin [4] Christensen and Irle [5] and so forthA very classicand successful model for population growth in mathematicsis logistic model
119889119883119905= (119903119883
119905minus 1198871198832
119905) 119889119905 (1)
where 119883119905denotes the density of resource population at time
119905 119903 gt 0 is called the intrinsic growth rate and 119887 = 119903119870 gt
0 (119870 is the environmental carrying capacity) The logisticmodel is used widely to real data however it is too simple toprovide a better simulation of the real world since there aresome uncertainties such as environment and financial effectmodeled by Gaussian white noise Hence the stochastic
logistic differential equation is introduced to handle theseproblems that is
119889119883119905= (119903119883
119905minus 1198871198832
119905) 119889119905 + 120583119883
119905119889119861119905
1198830= 119909 ge 0 119905 ge 0
(2)
where the constants 119903 119887 are mentioned in (1) 120583 is a measureof the size of the noise in the system and 119861
119905is 1-dimensional
Brownian motion defined on a complete probability space(ΩF F
119905119905ge0
P) satisfing the usual conditions There are somany extensive researches in literature such as Lungu andOslashksendal [6] Sun and Wang [7] Liu and Wang [8] and Liuand Wang [9 10]
Furthermore large and sudden fluctuations in environ-mental fluctuations can not modeled by the Gaussian whitenoise for examples hurricanes disasters and crashes APoisson jump stochastic equation can explain the suddenchanges In this paper we will concentrate on the stochasticlogistic population model with Poisson jump
119889119883119905= (119903119883
119905minus 1198871198832
119905) 119889119905 + 120583119883
119905119889119861119905
+ int
infin
0
119888119883119905minus (119889119905 119889119911) 119883
0= 119909 ge 0 119905 ge 0
(3)
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5839672 5 pageshttpdxdoiorg10115520165839672
2 Mathematical Problems in Engineering
where119883(119905minus) is the left limit of119883(119905) 119903 119887 120583 and 119861
119905are defined
in (2) 119888 is a bounded constant 119873 is a Poisson countingmeasurewith characteristicmeasure V on ameasurable subset119884 of (0infin) with V(119884) lt infin and (119889119905 119889119911) = 119873(119889119905 119889119911) minus
V(119889119911)119889119905 Throughout the paper we assume that 119861 and 119873
are independent More discussions of the stochastic jumpdiffusionmodel are given by Ryan and Hanson [11] Wee [12]Kunita [13] and Bao et al [14] and the references therein
Many methods such as Fokker-Planck equations timeaveraging methods and stochastic calculus are used onoptimal harvesting problems for model (2) all the aforemen-tioned works can be found in Alvarez and Shepp [15] Li andWang [16] and Li et al [17] To my best knowledge even formodel (2) there is little try by using optimal stopping theoryon optimal harvesting problems therefore in this paper wewill try the optimal stopping approach to solve the optimalharvesting problem for model (3) which is the motivation ofthe paper
The paper is organized as follows In Section 2 in orderto find the optimal value function and the optimal stoppingregion we formulate the problem and suppose we have a fishfactory with a population (eg a fish population in a pond)whose size 119883
119905at time 119905 is described by the stochastic jump
diffusion model (3) as a stopping problem In Section 3 anexplicit function for the value function is verifiedmeanwhilethe optimal stopping time and the optimal stopping regionare expressed
2 Description of Problem
Suppose the population with size 119883119905at time 119905 is given by the
stochastic logistic population model with Poisson jump
119889119883119905= (119903119883
119905minus 1198871198832
119905) 119889119905 + 120583119883
119905119889119861119905
+ int
infin
0
119888119883119905minus (119889119905 119889119911)
(4)
It can be proved that if 119903 gt 0 119887 gt 0 and 120583 119888 are boundedconstants then (4) has a unique positive solution 119883
119905defined
by
119883119909
119905=
Φ (119905)
1119909 + 119887 int119905
0Φ (119904) 119889119904
(5)
where
Φ (119905) = exp((119903 minus1
21205832) 119905 + 120583119861
119905
+ int119884
(ln (1 + 119888) minus 119888) 119905V (119889119911)
+ int
119905
0
int119884
ln (1 + 119888) (119889119904 119889119911))
(6)
for all 119905 ge 0 (see Bao et al [14]) and note that 0 le 119883119905lt 119870
Supposing that the population is say a fish populationin a pond the goal of this paper the optimal strategy for
selling a fish factory can be considered as an optimal stoppingproblem find Vlowast(119904 119909) and 120591
lowast such that
Vlowast (119904 119909) = sup120591
119864119909[119890minus120588120591
(119901119883120573
120591minus 119902)]
= 119864119909[119890minus120588120591lowast
(119901119883120573
120591lowast
minus 119902)] 120573 119901 119902 gt 0
(7)
the sup is taken over all stopping times 120591 of the process 119883119905
119905 gt 0 with the reward function
119877 (119904 119909) = 119890minus120588119904
(119901119909120573minus 119902) (8)
where the discounted exponent is 120588 gt 0 119890minus120588119904(119901119909120573 minus 119902) is theprofit of selling fish at time 120591 and 119902 represents a fixed fee andit is nature to assume that 119902 lt 119870 119864119909 denotes the expectationwith respect to the probability law 119875
119909 of the process119883119905 119905 ge 0
starting at1198830= 119909 gt 0
We will search for an optimal stopping time 120591lowast given in
(30) with the optimal stopping boundary 119909lowast from (23) on the
interval (0 119870) such that we can obtain the optimal profit Vlowastin (28) and the optimal stopping region 119860 in (29) Note thatit is trivial that the initial value 119909 le 119902 so we further assume119909 gt 119902
3 Analysis
For the jump diffusion logistic population model
119889119883119905= (119903119883
119905minus 1198871198832
119905) 119889119905 + 120583119883
119905119889119861119905
+ int
infin
0
119888119883119905minus (119889119905 119889119911)
(9)
and applying the Ito formula to a 1198622minusfunction 119891 such that
E[int119905
0int119884|119891(119905 119911)|V(119889119911)119889119905] lt infin and 119891
1015840 11989110158401015840 are bounded we
have the infinitesimal generator of the process 119891(119883119905) that is
[L119891] (119909)
= (119903119909 minus 1198871199092) 1198911015840(119909) +
1
21205832119909211989110158401015840(119909)
+ int119884
119891 (119909 + 119888119909) minus 119891 (119909) minus 1198881199091198911015840(119909) V (119889119911)
(10)
provided that int119884119891(119909 + 119888119909) minus 119891(119909) minus 119888119909119891
1015840(119909)V(119889119911) is well
defined since10038161003816100381610038161003816119891 (119909 + 119888119909) minus 119891 (119909) minus 119888119909119891
1015840(119909)
10038161003816100381610038161003816le
1
211988821199092 1003816100381610038161003816100381611989110158401015840(119909)
10038161003816100381610038161003816(11)
and 119888119909 are boundedNow let us consider a function equation
[L119891] (119909) = 0 119909 isin R+ (12)
We can try a solution of the form 119891(119909) = 120572119909120573 119909 isin R
+to
determine the unknown function that is
[L119891] (119909) = 120572119909120573[120573 (119903 minus 119887119909) +
1
2120573 (120573 minus 1) 120583
2
+ int119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)] ≜ 119891 (119909) 119892 (120573) = 0
(13)
Mathematical Problems in Engineering 3
where
119892 (120573) = 120573 (119903 minus 119887119909) +1
2120573 (120573 minus 1) 120583
2
+ int119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)
(14)
is well defined
Lemma 1 119892(120573) = 0 has two distinct real roots the largest one1205732 of which satisfies
0 lt 1205732lt 1 (15)
Proof The function 119892(120573) is decomposed into the sum of twofunctions
1198921(120573) = 120573 (
1
2(120573 minus 1) 120583
2+ 119903 minus 119887119909) forall119909 lt 119870
1198922(120573) = int
119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)
(16)
Since the former 1198921is a mixture of convex exponential
function (1+119888)120573 (119888 is bounded) we assume that119892(120573) is strictly
convex function Furthermore we have119892 (0) = 0
119892 (1) = 119887 (119870 minus 119909) gt 0
(17)
therefore the nonlinear equation 119892(120573) = 0 has two distinctreal roots 120573
1 1205732such that 120573
1= 0 and 0 lt 120573
2lt 1 respectively
We assume the following
Assumption 1
0 lt 120573 lt 1205732 (18)
Assumption 21
21205832119901120573 (120573 minus 1) + int
119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)
lt 0
(19)
Now let us define a function 119891lowast R+rarr R by
119891lowast(119909) =
119891 (119909) = 120572lowast1199091205732 0 lt 119909 lt 119909
lowast
119877 (119909) = 119901119909120573minus 119902 119909 ge 119909
lowast
(20)
where 120572lowast and 119909
lowastgt 0 are constants which are uniquely
determined by the following equations [2 18]Value matching condition
119891 (119909lowast) = 119877 (119909
lowast)
1198911015840(119909lowast) = 1198771015840(119909lowast)
(21)
That is
120572lowast=
119902120573
1205732minus 120573
(1199021205732
119901 (1205732minus 120573)
)
1205732
120573
119909lowast= (
1199021205732
119901 (1205732minus 120573)
)
1120573
(22)
Lemma 2 Under Assumptions 1 and 2 the function 119891lowast
R+rarr R satisfies the following properties (1)ndash(4)(1) For any 119909 isin R
+
119891lowast(119909) ge 119877 (119909) (23)
(2) 119891lowast(119909) is strictly increasing in 119909
(3) For any 119909 isin R+(119909 = 119909
lowast)
[L119891lowast] (119909) le 0 (24)
(4) For any 119909 isin R+ either ineq (23) or (24) holds with
equality
Proof(1) Setting 119910(119909) = 119891(119909) minus 119877(119909) = 120572
lowast1199091205732 minus 119901119909
120573+ 119902 and
differentiating 119910(119909) with respect to 119909 we have
1199101015840(119909) = 119891
1015840(119909) minus 119877
1015840(119909)
= 119909120573minus1
(120572lowast12057321199091205732
minus120573minus 119901120573)
lt 0 0 lt 119909 lt 119909lowast
ge 0 119909 ge 119909lowast
(25)
hence under Assumption 1 120572lowast12057321199091205732
minus120573minus 119901120573 is an increasing
function on [0infin) and we obtain the conclusion with thehelp of the fact that 119910(0) = 119902 119910(119909
lowast) = 0 119910(+infin) = +infin
(2) It is obvious(3) For 0 lt 119909 lt 119909
lowast 119891lowast(119909) = 119891(119909) we have
[L119891lowast] (119909) = [L119891] (119909) = 0 (26)
from (12)For 119909 gt 119909
lowast 119891lowast(119909) = 119877(119909) = 119901119909120573minus 119902 we obtain
[L119891] (119909) = [L119877] (119909) = 119909120573[119901120573 (119903 minus 119887119909)
+1
21205832119901120573 (120573 minus 1)
+ int119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)] = 119901120573119909
120573(119903
minus 119887119909) + 119909120573[1
21205832119901120573 (120573 minus 1)
+ int119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)] le 0
(27)
under Assumption 2 We finished the proof of (3)(4) It is trivial from (1) and (3)
Now let us give the main theorem
Theorem 3 Under Assumptions 1 and 2 the function 119891lowast
R+rarr 119877 is the optimal value function that is
Vlowast (119909) = 119891lowast(119909) 119909 isin 119877
+ (28)
Moreover the optimal stopping region 119860(sub R+) and the
optimal stopping time 120591lowast are given by the following
119860 = 119909 isin R+ 119891lowast(119909) = 119877 (119909) = [119909
lowast +infin (29)
120591lowast fl inf 119905 isin R
+ 119883lowast
119905isin 119860 (30)
4 Mathematical Problems in Engineering
Proof Using the function 119891lowast
R+
rarr R we define a newstochastic processM = 119872
119905 119905 isin 119877
+ by
119872119905= 119890minus120572119905
119891lowast(119883119909
119905) minus 119891lowast(119909) + int
119905
0
119890minus120572119904
[L119891lowast] (119883119909
119904) 119889119904
+ 119872119905
(31)
where
119872119905= int
119905
0
120583119883119909
119905119891lowast1015840
(119883119909
119905) 119889119861119905
+ int
119905
0
int119884
(119891lowast(119883119909
119905+ 119888119883119909
119905) minus 119891lowast(119883119909
119905)) (119889119905 119889119911)
(32)
is a continuous local martingale and by applying the Itoformula for the process 119890minus120572119905119891lowast(119883119909
119905) we obtain119872
119905= 0
Lemma 2 (4) implies
119890minus120572119905
119891lowast(119883119909
119905) le 119891lowast(119909) +
119905 (33)
with the help of the optimal sample theorem for martingalewe have for any stopping time 120591 for the process
119905119905ge0
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) + E
119905and120591 (34)
which can be written by
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) (35)
by noting the obvious fact E119905and120591
= 0
Taking lim inf119905rarrinfin
of both sides of (35) we have by Fatoulemma
E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
] le 119891lowast(119909) (36)
moreover since the function 119891lowast has property Lemma 2 (1) it
holds that
E [119890minus120572120591
119877 (119883119909
119905) 119868120591ltinfin
] le E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
]
le 119891lowast(119909)
(37)
On the other hand for the stopping time 120591lowast defined by
(30)
E [119890minus120572(120591lowast
and119905)119891lowast(119883119909
120591lowast
and119905)] = 119891
lowast(119909) (38)
By the properties of Lemma 2 (1)ndash(4) of the function 119891lowast
we assure that119860 = [119909lowastinfin) and by the properties of Lemma 2
(2) it holds that
0 lt 119891lowast(119883119909
120591lowast
and119905) le 119891lowast(119909lowast) as (39)
Taking lim119905isininfin
of both sides of (38) we have by thebounded convergence theorem of Lebesgue
119891lowast(119909) = E [119890
minus120572120591lowast
119891lowast(119883119909
120591lowast
) 119868120591lowast
ltinfin]
= E [119890minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin]
(40)
where the second equality follows from the fact that on theevent 120591lowast lt infin
119891lowast(119883119909
120591lowast
) = 119877 (119883119909
120591lowast
) (41)
Then we conclude that
119891lowastle E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] le 119891lowast (42)
that is
Vlowast (119909) = 119891lowast(119909) = E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] (43)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Yang Sun is supported by the NSFC Grant (no 51406044and no 11401085) and Natural Science Foundation of theEducation Department of Heilongjiang Province (Grant no12521116) Xiaohui Ai is supported by the NSFC Grant (no11401085) and the Fundamental Research Funds for theCentral Universities (no 2572015BB14)
References
[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewal Resources John Wiley amp Sons New York NYUSA 2nd edition 1990
[2] S Dayanik and I Karatzas ldquoOn the optimal stopping problemfor one-dimensional diffusionsrdquo Stochastic Processes and TheirApplications vol 107 no 2 pp 173ndash212 2003
[3] M Dai and Y K Kwok ldquoCharacterization of optimal stoppingregions of American Asian and lookback optionsrdquoMathemati-cal Finance vol 16 no 1 pp 63ndash82 2006
[4] E L Presman and I M Sonin ldquoOn an optimal stopping prob-lem for random variables defined on aMarkov chainrdquoTheory ofProbability amp Its Applications vol 54 no 3 pp 599ndash608 2009
[5] S Christensen and A Irle ldquoA harmonic function technique forthe optimal stopping of diffusionsrdquo Stochastics vol 83 no 4ndash6pp 347ndash363 2011
[6] E M Lungu and B Oslashksendal ldquoOptimal harvesting from a pop-ulation in a stochastic crowded environmentrdquo MathematicalBiosciences vol 145 no 1 pp 47ndash75 1997
[7] X Sun and Y Wang ldquoStability analysis of a stochastic logisticmodel with nonlinear diffusion termrdquo Applied MathematicalModelling vol 32 no 10 pp 2067ndash2075 2008
[8] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[9] M Liu and K Wang ldquoStationary distribution ergodicity andextinction of a stochastic generalized logistic systemrdquo AppliedMathematics Letters vol 25 no 11 pp 1980ndash1985 2012
[10] M Liu and K Wang ldquoOn a stochastic logistic equation withimpulsive perturbationsrdquoComputersampMathematics withAppli-cations vol 63 no 5 pp 871ndash886 2012
Mathematical Problems in Engineering 5
[11] D Ryan and F B Hanson ldquoOptimal harvesting of a logisticpopulation in an environment with stochastic jumpsrdquo Journalof Mathematical Biology vol 24 no 3 pp 259ndash277 1986
[12] I-S Wee ldquoStability for multidimensional jump-diffusion pro-cessesrdquo Stochastic Processes and Their Applications vol 80 no2 pp 193ndash209 1999
[13] H Kunita ldquoItorsquos stochastic calculus its surprising power forapplicationsrdquo Stochastic Processes and Their Applications vol120 no 5 pp 622ndash652 2010
[14] J Bao X Mao G Yin and C Yuan ldquoCompetitive Lotka-Volterra population dynamics with jumpsrdquo Nonlinear AnalysisTheory Methods amp Applications Series A Theory and Methodsvol 74 no 17 pp 6601ndash6616 2011
[15] L H R Alvarez and L A Shepp ldquoOptimal harvesting ofstochastically fluctuating populationsrdquo Journal of MathematicalBiology vol 37 no 2 pp 155ndash177 1998
[16] W Li and K Wang ldquoOptimal harvesting policy for generalstochastic logistic population modelrdquo Journal of MathematicalAnalysis and Applications vol 368 no 2 pp 420ndash428 2010
[17] W Li K Wang and H Su ldquoOptimal harvesting policy forstochastic Logistic populationmodelrdquoAppliedMathematics andComputation vol 218 no 1 pp 157ndash162 2011
[18] A Dixit The Art of Smooth Pasting Harwood Academic Pub-lishers Chur Switzerland 1993
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2 Mathematical Problems in Engineering
where119883(119905minus) is the left limit of119883(119905) 119903 119887 120583 and 119861
119905are defined
in (2) 119888 is a bounded constant 119873 is a Poisson countingmeasurewith characteristicmeasure V on ameasurable subset119884 of (0infin) with V(119884) lt infin and (119889119905 119889119911) = 119873(119889119905 119889119911) minus
V(119889119911)119889119905 Throughout the paper we assume that 119861 and 119873
are independent More discussions of the stochastic jumpdiffusionmodel are given by Ryan and Hanson [11] Wee [12]Kunita [13] and Bao et al [14] and the references therein
Many methods such as Fokker-Planck equations timeaveraging methods and stochastic calculus are used onoptimal harvesting problems for model (2) all the aforemen-tioned works can be found in Alvarez and Shepp [15] Li andWang [16] and Li et al [17] To my best knowledge even formodel (2) there is little try by using optimal stopping theoryon optimal harvesting problems therefore in this paper wewill try the optimal stopping approach to solve the optimalharvesting problem for model (3) which is the motivation ofthe paper
The paper is organized as follows In Section 2 in orderto find the optimal value function and the optimal stoppingregion we formulate the problem and suppose we have a fishfactory with a population (eg a fish population in a pond)whose size 119883
119905at time 119905 is described by the stochastic jump
diffusion model (3) as a stopping problem In Section 3 anexplicit function for the value function is verifiedmeanwhilethe optimal stopping time and the optimal stopping regionare expressed
2 Description of Problem
Suppose the population with size 119883119905at time 119905 is given by the
stochastic logistic population model with Poisson jump
119889119883119905= (119903119883
119905minus 1198871198832
119905) 119889119905 + 120583119883
119905119889119861119905
+ int
infin
0
119888119883119905minus (119889119905 119889119911)
(4)
It can be proved that if 119903 gt 0 119887 gt 0 and 120583 119888 are boundedconstants then (4) has a unique positive solution 119883
119905defined
by
119883119909
119905=
Φ (119905)
1119909 + 119887 int119905
0Φ (119904) 119889119904
(5)
where
Φ (119905) = exp((119903 minus1
21205832) 119905 + 120583119861
119905
+ int119884
(ln (1 + 119888) minus 119888) 119905V (119889119911)
+ int
119905
0
int119884
ln (1 + 119888) (119889119904 119889119911))
(6)
for all 119905 ge 0 (see Bao et al [14]) and note that 0 le 119883119905lt 119870
Supposing that the population is say a fish populationin a pond the goal of this paper the optimal strategy for
selling a fish factory can be considered as an optimal stoppingproblem find Vlowast(119904 119909) and 120591
lowast such that
Vlowast (119904 119909) = sup120591
119864119909[119890minus120588120591
(119901119883120573
120591minus 119902)]
= 119864119909[119890minus120588120591lowast
(119901119883120573
120591lowast
minus 119902)] 120573 119901 119902 gt 0
(7)
the sup is taken over all stopping times 120591 of the process 119883119905
119905 gt 0 with the reward function
119877 (119904 119909) = 119890minus120588119904
(119901119909120573minus 119902) (8)
where the discounted exponent is 120588 gt 0 119890minus120588119904(119901119909120573 minus 119902) is theprofit of selling fish at time 120591 and 119902 represents a fixed fee andit is nature to assume that 119902 lt 119870 119864119909 denotes the expectationwith respect to the probability law 119875
119909 of the process119883119905 119905 ge 0
starting at1198830= 119909 gt 0
We will search for an optimal stopping time 120591lowast given in
(30) with the optimal stopping boundary 119909lowast from (23) on the
interval (0 119870) such that we can obtain the optimal profit Vlowastin (28) and the optimal stopping region 119860 in (29) Note thatit is trivial that the initial value 119909 le 119902 so we further assume119909 gt 119902
3 Analysis
For the jump diffusion logistic population model
119889119883119905= (119903119883
119905minus 1198871198832
119905) 119889119905 + 120583119883
119905119889119861119905
+ int
infin
0
119888119883119905minus (119889119905 119889119911)
(9)
and applying the Ito formula to a 1198622minusfunction 119891 such that
E[int119905
0int119884|119891(119905 119911)|V(119889119911)119889119905] lt infin and 119891
1015840 11989110158401015840 are bounded we
have the infinitesimal generator of the process 119891(119883119905) that is
[L119891] (119909)
= (119903119909 minus 1198871199092) 1198911015840(119909) +
1
21205832119909211989110158401015840(119909)
+ int119884
119891 (119909 + 119888119909) minus 119891 (119909) minus 1198881199091198911015840(119909) V (119889119911)
(10)
provided that int119884119891(119909 + 119888119909) minus 119891(119909) minus 119888119909119891
1015840(119909)V(119889119911) is well
defined since10038161003816100381610038161003816119891 (119909 + 119888119909) minus 119891 (119909) minus 119888119909119891
1015840(119909)
10038161003816100381610038161003816le
1
211988821199092 1003816100381610038161003816100381611989110158401015840(119909)
10038161003816100381610038161003816(11)
and 119888119909 are boundedNow let us consider a function equation
[L119891] (119909) = 0 119909 isin R+ (12)
We can try a solution of the form 119891(119909) = 120572119909120573 119909 isin R
+to
determine the unknown function that is
[L119891] (119909) = 120572119909120573[120573 (119903 minus 119887119909) +
1
2120573 (120573 minus 1) 120583
2
+ int119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)] ≜ 119891 (119909) 119892 (120573) = 0
(13)
Mathematical Problems in Engineering 3
where
119892 (120573) = 120573 (119903 minus 119887119909) +1
2120573 (120573 minus 1) 120583
2
+ int119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)
(14)
is well defined
Lemma 1 119892(120573) = 0 has two distinct real roots the largest one1205732 of which satisfies
0 lt 1205732lt 1 (15)
Proof The function 119892(120573) is decomposed into the sum of twofunctions
1198921(120573) = 120573 (
1
2(120573 minus 1) 120583
2+ 119903 minus 119887119909) forall119909 lt 119870
1198922(120573) = int
119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)
(16)
Since the former 1198921is a mixture of convex exponential
function (1+119888)120573 (119888 is bounded) we assume that119892(120573) is strictly
convex function Furthermore we have119892 (0) = 0
119892 (1) = 119887 (119870 minus 119909) gt 0
(17)
therefore the nonlinear equation 119892(120573) = 0 has two distinctreal roots 120573
1 1205732such that 120573
1= 0 and 0 lt 120573
2lt 1 respectively
We assume the following
Assumption 1
0 lt 120573 lt 1205732 (18)
Assumption 21
21205832119901120573 (120573 minus 1) + int
119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)
lt 0
(19)
Now let us define a function 119891lowast R+rarr R by
119891lowast(119909) =
119891 (119909) = 120572lowast1199091205732 0 lt 119909 lt 119909
lowast
119877 (119909) = 119901119909120573minus 119902 119909 ge 119909
lowast
(20)
where 120572lowast and 119909
lowastgt 0 are constants which are uniquely
determined by the following equations [2 18]Value matching condition
119891 (119909lowast) = 119877 (119909
lowast)
1198911015840(119909lowast) = 1198771015840(119909lowast)
(21)
That is
120572lowast=
119902120573
1205732minus 120573
(1199021205732
119901 (1205732minus 120573)
)
1205732
120573
119909lowast= (
1199021205732
119901 (1205732minus 120573)
)
1120573
(22)
Lemma 2 Under Assumptions 1 and 2 the function 119891lowast
R+rarr R satisfies the following properties (1)ndash(4)(1) For any 119909 isin R
+
119891lowast(119909) ge 119877 (119909) (23)
(2) 119891lowast(119909) is strictly increasing in 119909
(3) For any 119909 isin R+(119909 = 119909
lowast)
[L119891lowast] (119909) le 0 (24)
(4) For any 119909 isin R+ either ineq (23) or (24) holds with
equality
Proof(1) Setting 119910(119909) = 119891(119909) minus 119877(119909) = 120572
lowast1199091205732 minus 119901119909
120573+ 119902 and
differentiating 119910(119909) with respect to 119909 we have
1199101015840(119909) = 119891
1015840(119909) minus 119877
1015840(119909)
= 119909120573minus1
(120572lowast12057321199091205732
minus120573minus 119901120573)
lt 0 0 lt 119909 lt 119909lowast
ge 0 119909 ge 119909lowast
(25)
hence under Assumption 1 120572lowast12057321199091205732
minus120573minus 119901120573 is an increasing
function on [0infin) and we obtain the conclusion with thehelp of the fact that 119910(0) = 119902 119910(119909
lowast) = 0 119910(+infin) = +infin
(2) It is obvious(3) For 0 lt 119909 lt 119909
lowast 119891lowast(119909) = 119891(119909) we have
[L119891lowast] (119909) = [L119891] (119909) = 0 (26)
from (12)For 119909 gt 119909
lowast 119891lowast(119909) = 119877(119909) = 119901119909120573minus 119902 we obtain
[L119891] (119909) = [L119877] (119909) = 119909120573[119901120573 (119903 minus 119887119909)
+1
21205832119901120573 (120573 minus 1)
+ int119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)] = 119901120573119909
120573(119903
minus 119887119909) + 119909120573[1
21205832119901120573 (120573 minus 1)
+ int119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)] le 0
(27)
under Assumption 2 We finished the proof of (3)(4) It is trivial from (1) and (3)
Now let us give the main theorem
Theorem 3 Under Assumptions 1 and 2 the function 119891lowast
R+rarr 119877 is the optimal value function that is
Vlowast (119909) = 119891lowast(119909) 119909 isin 119877
+ (28)
Moreover the optimal stopping region 119860(sub R+) and the
optimal stopping time 120591lowast are given by the following
119860 = 119909 isin R+ 119891lowast(119909) = 119877 (119909) = [119909
lowast +infin (29)
120591lowast fl inf 119905 isin R
+ 119883lowast
119905isin 119860 (30)
4 Mathematical Problems in Engineering
Proof Using the function 119891lowast
R+
rarr R we define a newstochastic processM = 119872
119905 119905 isin 119877
+ by
119872119905= 119890minus120572119905
119891lowast(119883119909
119905) minus 119891lowast(119909) + int
119905
0
119890minus120572119904
[L119891lowast] (119883119909
119904) 119889119904
+ 119872119905
(31)
where
119872119905= int
119905
0
120583119883119909
119905119891lowast1015840
(119883119909
119905) 119889119861119905
+ int
119905
0
int119884
(119891lowast(119883119909
119905+ 119888119883119909
119905) minus 119891lowast(119883119909
119905)) (119889119905 119889119911)
(32)
is a continuous local martingale and by applying the Itoformula for the process 119890minus120572119905119891lowast(119883119909
119905) we obtain119872
119905= 0
Lemma 2 (4) implies
119890minus120572119905
119891lowast(119883119909
119905) le 119891lowast(119909) +
119905 (33)
with the help of the optimal sample theorem for martingalewe have for any stopping time 120591 for the process
119905119905ge0
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) + E
119905and120591 (34)
which can be written by
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) (35)
by noting the obvious fact E119905and120591
= 0
Taking lim inf119905rarrinfin
of both sides of (35) we have by Fatoulemma
E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
] le 119891lowast(119909) (36)
moreover since the function 119891lowast has property Lemma 2 (1) it
holds that
E [119890minus120572120591
119877 (119883119909
119905) 119868120591ltinfin
] le E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
]
le 119891lowast(119909)
(37)
On the other hand for the stopping time 120591lowast defined by
(30)
E [119890minus120572(120591lowast
and119905)119891lowast(119883119909
120591lowast
and119905)] = 119891
lowast(119909) (38)
By the properties of Lemma 2 (1)ndash(4) of the function 119891lowast
we assure that119860 = [119909lowastinfin) and by the properties of Lemma 2
(2) it holds that
0 lt 119891lowast(119883119909
120591lowast
and119905) le 119891lowast(119909lowast) as (39)
Taking lim119905isininfin
of both sides of (38) we have by thebounded convergence theorem of Lebesgue
119891lowast(119909) = E [119890
minus120572120591lowast
119891lowast(119883119909
120591lowast
) 119868120591lowast
ltinfin]
= E [119890minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin]
(40)
where the second equality follows from the fact that on theevent 120591lowast lt infin
119891lowast(119883119909
120591lowast
) = 119877 (119883119909
120591lowast
) (41)
Then we conclude that
119891lowastle E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] le 119891lowast (42)
that is
Vlowast (119909) = 119891lowast(119909) = E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] (43)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Yang Sun is supported by the NSFC Grant (no 51406044and no 11401085) and Natural Science Foundation of theEducation Department of Heilongjiang Province (Grant no12521116) Xiaohui Ai is supported by the NSFC Grant (no11401085) and the Fundamental Research Funds for theCentral Universities (no 2572015BB14)
References
[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewal Resources John Wiley amp Sons New York NYUSA 2nd edition 1990
[2] S Dayanik and I Karatzas ldquoOn the optimal stopping problemfor one-dimensional diffusionsrdquo Stochastic Processes and TheirApplications vol 107 no 2 pp 173ndash212 2003
[3] M Dai and Y K Kwok ldquoCharacterization of optimal stoppingregions of American Asian and lookback optionsrdquoMathemati-cal Finance vol 16 no 1 pp 63ndash82 2006
[4] E L Presman and I M Sonin ldquoOn an optimal stopping prob-lem for random variables defined on aMarkov chainrdquoTheory ofProbability amp Its Applications vol 54 no 3 pp 599ndash608 2009
[5] S Christensen and A Irle ldquoA harmonic function technique forthe optimal stopping of diffusionsrdquo Stochastics vol 83 no 4ndash6pp 347ndash363 2011
[6] E M Lungu and B Oslashksendal ldquoOptimal harvesting from a pop-ulation in a stochastic crowded environmentrdquo MathematicalBiosciences vol 145 no 1 pp 47ndash75 1997
[7] X Sun and Y Wang ldquoStability analysis of a stochastic logisticmodel with nonlinear diffusion termrdquo Applied MathematicalModelling vol 32 no 10 pp 2067ndash2075 2008
[8] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[9] M Liu and K Wang ldquoStationary distribution ergodicity andextinction of a stochastic generalized logistic systemrdquo AppliedMathematics Letters vol 25 no 11 pp 1980ndash1985 2012
[10] M Liu and K Wang ldquoOn a stochastic logistic equation withimpulsive perturbationsrdquoComputersampMathematics withAppli-cations vol 63 no 5 pp 871ndash886 2012
Mathematical Problems in Engineering 5
[11] D Ryan and F B Hanson ldquoOptimal harvesting of a logisticpopulation in an environment with stochastic jumpsrdquo Journalof Mathematical Biology vol 24 no 3 pp 259ndash277 1986
[12] I-S Wee ldquoStability for multidimensional jump-diffusion pro-cessesrdquo Stochastic Processes and Their Applications vol 80 no2 pp 193ndash209 1999
[13] H Kunita ldquoItorsquos stochastic calculus its surprising power forapplicationsrdquo Stochastic Processes and Their Applications vol120 no 5 pp 622ndash652 2010
[14] J Bao X Mao G Yin and C Yuan ldquoCompetitive Lotka-Volterra population dynamics with jumpsrdquo Nonlinear AnalysisTheory Methods amp Applications Series A Theory and Methodsvol 74 no 17 pp 6601ndash6616 2011
[15] L H R Alvarez and L A Shepp ldquoOptimal harvesting ofstochastically fluctuating populationsrdquo Journal of MathematicalBiology vol 37 no 2 pp 155ndash177 1998
[16] W Li and K Wang ldquoOptimal harvesting policy for generalstochastic logistic population modelrdquo Journal of MathematicalAnalysis and Applications vol 368 no 2 pp 420ndash428 2010
[17] W Li K Wang and H Su ldquoOptimal harvesting policy forstochastic Logistic populationmodelrdquoAppliedMathematics andComputation vol 218 no 1 pp 157ndash162 2011
[18] A Dixit The Art of Smooth Pasting Harwood Academic Pub-lishers Chur Switzerland 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
where
119892 (120573) = 120573 (119903 minus 119887119909) +1
2120573 (120573 minus 1) 120583
2
+ int119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)
(14)
is well defined
Lemma 1 119892(120573) = 0 has two distinct real roots the largest one1205732 of which satisfies
0 lt 1205732lt 1 (15)
Proof The function 119892(120573) is decomposed into the sum of twofunctions
1198921(120573) = 120573 (
1
2(120573 minus 1) 120583
2+ 119903 minus 119887119909) forall119909 lt 119870
1198922(120573) = int
119884
(1 + 119888)120573minus 1 minus 120573119888 V (119889119911)
(16)
Since the former 1198921is a mixture of convex exponential
function (1+119888)120573 (119888 is bounded) we assume that119892(120573) is strictly
convex function Furthermore we have119892 (0) = 0
119892 (1) = 119887 (119870 minus 119909) gt 0
(17)
therefore the nonlinear equation 119892(120573) = 0 has two distinctreal roots 120573
1 1205732such that 120573
1= 0 and 0 lt 120573
2lt 1 respectively
We assume the following
Assumption 1
0 lt 120573 lt 1205732 (18)
Assumption 21
21205832119901120573 (120573 minus 1) + int
119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)
lt 0
(19)
Now let us define a function 119891lowast R+rarr R by
119891lowast(119909) =
119891 (119909) = 120572lowast1199091205732 0 lt 119909 lt 119909
lowast
119877 (119909) = 119901119909120573minus 119902 119909 ge 119909
lowast
(20)
where 120572lowast and 119909
lowastgt 0 are constants which are uniquely
determined by the following equations [2 18]Value matching condition
119891 (119909lowast) = 119877 (119909
lowast)
1198911015840(119909lowast) = 1198771015840(119909lowast)
(21)
That is
120572lowast=
119902120573
1205732minus 120573
(1199021205732
119901 (1205732minus 120573)
)
1205732
120573
119909lowast= (
1199021205732
119901 (1205732minus 120573)
)
1120573
(22)
Lemma 2 Under Assumptions 1 and 2 the function 119891lowast
R+rarr R satisfies the following properties (1)ndash(4)(1) For any 119909 isin R
+
119891lowast(119909) ge 119877 (119909) (23)
(2) 119891lowast(119909) is strictly increasing in 119909
(3) For any 119909 isin R+(119909 = 119909
lowast)
[L119891lowast] (119909) le 0 (24)
(4) For any 119909 isin R+ either ineq (23) or (24) holds with
equality
Proof(1) Setting 119910(119909) = 119891(119909) minus 119877(119909) = 120572
lowast1199091205732 minus 119901119909
120573+ 119902 and
differentiating 119910(119909) with respect to 119909 we have
1199101015840(119909) = 119891
1015840(119909) minus 119877
1015840(119909)
= 119909120573minus1
(120572lowast12057321199091205732
minus120573minus 119901120573)
lt 0 0 lt 119909 lt 119909lowast
ge 0 119909 ge 119909lowast
(25)
hence under Assumption 1 120572lowast12057321199091205732
minus120573minus 119901120573 is an increasing
function on [0infin) and we obtain the conclusion with thehelp of the fact that 119910(0) = 119902 119910(119909
lowast) = 0 119910(+infin) = +infin
(2) It is obvious(3) For 0 lt 119909 lt 119909
lowast 119891lowast(119909) = 119891(119909) we have
[L119891lowast] (119909) = [L119891] (119909) = 0 (26)
from (12)For 119909 gt 119909
lowast 119891lowast(119909) = 119877(119909) = 119901119909120573minus 119902 we obtain
[L119891] (119909) = [L119877] (119909) = 119909120573[119901120573 (119903 minus 119887119909)
+1
21205832119901120573 (120573 minus 1)
+ int119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)] = 119901120573119909
120573(119903
minus 119887119909) + 119909120573[1
21205832119901120573 (120573 minus 1)
+ int119884
(119901 (1 + 119888)120573minus 119901 minus 119888119901120573) V (119889119911)] le 0
(27)
under Assumption 2 We finished the proof of (3)(4) It is trivial from (1) and (3)
Now let us give the main theorem
Theorem 3 Under Assumptions 1 and 2 the function 119891lowast
R+rarr 119877 is the optimal value function that is
Vlowast (119909) = 119891lowast(119909) 119909 isin 119877
+ (28)
Moreover the optimal stopping region 119860(sub R+) and the
optimal stopping time 120591lowast are given by the following
119860 = 119909 isin R+ 119891lowast(119909) = 119877 (119909) = [119909
lowast +infin (29)
120591lowast fl inf 119905 isin R
+ 119883lowast
119905isin 119860 (30)
4 Mathematical Problems in Engineering
Proof Using the function 119891lowast
R+
rarr R we define a newstochastic processM = 119872
119905 119905 isin 119877
+ by
119872119905= 119890minus120572119905
119891lowast(119883119909
119905) minus 119891lowast(119909) + int
119905
0
119890minus120572119904
[L119891lowast] (119883119909
119904) 119889119904
+ 119872119905
(31)
where
119872119905= int
119905
0
120583119883119909
119905119891lowast1015840
(119883119909
119905) 119889119861119905
+ int
119905
0
int119884
(119891lowast(119883119909
119905+ 119888119883119909
119905) minus 119891lowast(119883119909
119905)) (119889119905 119889119911)
(32)
is a continuous local martingale and by applying the Itoformula for the process 119890minus120572119905119891lowast(119883119909
119905) we obtain119872
119905= 0
Lemma 2 (4) implies
119890minus120572119905
119891lowast(119883119909
119905) le 119891lowast(119909) +
119905 (33)
with the help of the optimal sample theorem for martingalewe have for any stopping time 120591 for the process
119905119905ge0
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) + E
119905and120591 (34)
which can be written by
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) (35)
by noting the obvious fact E119905and120591
= 0
Taking lim inf119905rarrinfin
of both sides of (35) we have by Fatoulemma
E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
] le 119891lowast(119909) (36)
moreover since the function 119891lowast has property Lemma 2 (1) it
holds that
E [119890minus120572120591
119877 (119883119909
119905) 119868120591ltinfin
] le E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
]
le 119891lowast(119909)
(37)
On the other hand for the stopping time 120591lowast defined by
(30)
E [119890minus120572(120591lowast
and119905)119891lowast(119883119909
120591lowast
and119905)] = 119891
lowast(119909) (38)
By the properties of Lemma 2 (1)ndash(4) of the function 119891lowast
we assure that119860 = [119909lowastinfin) and by the properties of Lemma 2
(2) it holds that
0 lt 119891lowast(119883119909
120591lowast
and119905) le 119891lowast(119909lowast) as (39)
Taking lim119905isininfin
of both sides of (38) we have by thebounded convergence theorem of Lebesgue
119891lowast(119909) = E [119890
minus120572120591lowast
119891lowast(119883119909
120591lowast
) 119868120591lowast
ltinfin]
= E [119890minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin]
(40)
where the second equality follows from the fact that on theevent 120591lowast lt infin
119891lowast(119883119909
120591lowast
) = 119877 (119883119909
120591lowast
) (41)
Then we conclude that
119891lowastle E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] le 119891lowast (42)
that is
Vlowast (119909) = 119891lowast(119909) = E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] (43)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Yang Sun is supported by the NSFC Grant (no 51406044and no 11401085) and Natural Science Foundation of theEducation Department of Heilongjiang Province (Grant no12521116) Xiaohui Ai is supported by the NSFC Grant (no11401085) and the Fundamental Research Funds for theCentral Universities (no 2572015BB14)
References
[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewal Resources John Wiley amp Sons New York NYUSA 2nd edition 1990
[2] S Dayanik and I Karatzas ldquoOn the optimal stopping problemfor one-dimensional diffusionsrdquo Stochastic Processes and TheirApplications vol 107 no 2 pp 173ndash212 2003
[3] M Dai and Y K Kwok ldquoCharacterization of optimal stoppingregions of American Asian and lookback optionsrdquoMathemati-cal Finance vol 16 no 1 pp 63ndash82 2006
[4] E L Presman and I M Sonin ldquoOn an optimal stopping prob-lem for random variables defined on aMarkov chainrdquoTheory ofProbability amp Its Applications vol 54 no 3 pp 599ndash608 2009
[5] S Christensen and A Irle ldquoA harmonic function technique forthe optimal stopping of diffusionsrdquo Stochastics vol 83 no 4ndash6pp 347ndash363 2011
[6] E M Lungu and B Oslashksendal ldquoOptimal harvesting from a pop-ulation in a stochastic crowded environmentrdquo MathematicalBiosciences vol 145 no 1 pp 47ndash75 1997
[7] X Sun and Y Wang ldquoStability analysis of a stochastic logisticmodel with nonlinear diffusion termrdquo Applied MathematicalModelling vol 32 no 10 pp 2067ndash2075 2008
[8] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[9] M Liu and K Wang ldquoStationary distribution ergodicity andextinction of a stochastic generalized logistic systemrdquo AppliedMathematics Letters vol 25 no 11 pp 1980ndash1985 2012
[10] M Liu and K Wang ldquoOn a stochastic logistic equation withimpulsive perturbationsrdquoComputersampMathematics withAppli-cations vol 63 no 5 pp 871ndash886 2012
Mathematical Problems in Engineering 5
[11] D Ryan and F B Hanson ldquoOptimal harvesting of a logisticpopulation in an environment with stochastic jumpsrdquo Journalof Mathematical Biology vol 24 no 3 pp 259ndash277 1986
[12] I-S Wee ldquoStability for multidimensional jump-diffusion pro-cessesrdquo Stochastic Processes and Their Applications vol 80 no2 pp 193ndash209 1999
[13] H Kunita ldquoItorsquos stochastic calculus its surprising power forapplicationsrdquo Stochastic Processes and Their Applications vol120 no 5 pp 622ndash652 2010
[14] J Bao X Mao G Yin and C Yuan ldquoCompetitive Lotka-Volterra population dynamics with jumpsrdquo Nonlinear AnalysisTheory Methods amp Applications Series A Theory and Methodsvol 74 no 17 pp 6601ndash6616 2011
[15] L H R Alvarez and L A Shepp ldquoOptimal harvesting ofstochastically fluctuating populationsrdquo Journal of MathematicalBiology vol 37 no 2 pp 155ndash177 1998
[16] W Li and K Wang ldquoOptimal harvesting policy for generalstochastic logistic population modelrdquo Journal of MathematicalAnalysis and Applications vol 368 no 2 pp 420ndash428 2010
[17] W Li K Wang and H Su ldquoOptimal harvesting policy forstochastic Logistic populationmodelrdquoAppliedMathematics andComputation vol 218 no 1 pp 157ndash162 2011
[18] A Dixit The Art of Smooth Pasting Harwood Academic Pub-lishers Chur Switzerland 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Proof Using the function 119891lowast
R+
rarr R we define a newstochastic processM = 119872
119905 119905 isin 119877
+ by
119872119905= 119890minus120572119905
119891lowast(119883119909
119905) minus 119891lowast(119909) + int
119905
0
119890minus120572119904
[L119891lowast] (119883119909
119904) 119889119904
+ 119872119905
(31)
where
119872119905= int
119905
0
120583119883119909
119905119891lowast1015840
(119883119909
119905) 119889119861119905
+ int
119905
0
int119884
(119891lowast(119883119909
119905+ 119888119883119909
119905) minus 119891lowast(119883119909
119905)) (119889119905 119889119911)
(32)
is a continuous local martingale and by applying the Itoformula for the process 119890minus120572119905119891lowast(119883119909
119905) we obtain119872
119905= 0
Lemma 2 (4) implies
119890minus120572119905
119891lowast(119883119909
119905) le 119891lowast(119909) +
119905 (33)
with the help of the optimal sample theorem for martingalewe have for any stopping time 120591 for the process
119905119905ge0
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) + E
119905and120591 (34)
which can be written by
E119890minus120572(119905and120591)
119891lowast(119883119909
119905and120591) le 119891lowast(119909) (35)
by noting the obvious fact E119905and120591
= 0
Taking lim inf119905rarrinfin
of both sides of (35) we have by Fatoulemma
E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
] le 119891lowast(119909) (36)
moreover since the function 119891lowast has property Lemma 2 (1) it
holds that
E [119890minus120572120591
119877 (119883119909
119905) 119868120591ltinfin
] le E [119890minus120572120591
119891lowast(119883119909
119905) 119868120591ltinfin
]
le 119891lowast(119909)
(37)
On the other hand for the stopping time 120591lowast defined by
(30)
E [119890minus120572(120591lowast
and119905)119891lowast(119883119909
120591lowast
and119905)] = 119891
lowast(119909) (38)
By the properties of Lemma 2 (1)ndash(4) of the function 119891lowast
we assure that119860 = [119909lowastinfin) and by the properties of Lemma 2
(2) it holds that
0 lt 119891lowast(119883119909
120591lowast
and119905) le 119891lowast(119909lowast) as (39)
Taking lim119905isininfin
of both sides of (38) we have by thebounded convergence theorem of Lebesgue
119891lowast(119909) = E [119890
minus120572120591lowast
119891lowast(119883119909
120591lowast
) 119868120591lowast
ltinfin]
= E [119890minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin]
(40)
where the second equality follows from the fact that on theevent 120591lowast lt infin
119891lowast(119883119909
120591lowast
) = 119877 (119883119909
120591lowast
) (41)
Then we conclude that
119891lowastle E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] le 119891lowast (42)
that is
Vlowast (119909) = 119891lowast(119909) = E [119890
minus120572120591lowast
119877 (119883119909
120591lowast
) 119868120591lowast
ltinfin] (43)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Yang Sun is supported by the NSFC Grant (no 51406044and no 11401085) and Natural Science Foundation of theEducation Department of Heilongjiang Province (Grant no12521116) Xiaohui Ai is supported by the NSFC Grant (no11401085) and the Fundamental Research Funds for theCentral Universities (no 2572015BB14)
References
[1] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewal Resources John Wiley amp Sons New York NYUSA 2nd edition 1990
[2] S Dayanik and I Karatzas ldquoOn the optimal stopping problemfor one-dimensional diffusionsrdquo Stochastic Processes and TheirApplications vol 107 no 2 pp 173ndash212 2003
[3] M Dai and Y K Kwok ldquoCharacterization of optimal stoppingregions of American Asian and lookback optionsrdquoMathemati-cal Finance vol 16 no 1 pp 63ndash82 2006
[4] E L Presman and I M Sonin ldquoOn an optimal stopping prob-lem for random variables defined on aMarkov chainrdquoTheory ofProbability amp Its Applications vol 54 no 3 pp 599ndash608 2009
[5] S Christensen and A Irle ldquoA harmonic function technique forthe optimal stopping of diffusionsrdquo Stochastics vol 83 no 4ndash6pp 347ndash363 2011
[6] E M Lungu and B Oslashksendal ldquoOptimal harvesting from a pop-ulation in a stochastic crowded environmentrdquo MathematicalBiosciences vol 145 no 1 pp 47ndash75 1997
[7] X Sun and Y Wang ldquoStability analysis of a stochastic logisticmodel with nonlinear diffusion termrdquo Applied MathematicalModelling vol 32 no 10 pp 2067ndash2075 2008
[8] M Liu and KWang ldquoAsymptotic properties and simulations ofa stochastic logistic model under regime switchingrdquoMathemat-ical and Computer Modelling vol 54 no 9-10 pp 2139ndash21542011
[9] M Liu and K Wang ldquoStationary distribution ergodicity andextinction of a stochastic generalized logistic systemrdquo AppliedMathematics Letters vol 25 no 11 pp 1980ndash1985 2012
[10] M Liu and K Wang ldquoOn a stochastic logistic equation withimpulsive perturbationsrdquoComputersampMathematics withAppli-cations vol 63 no 5 pp 871ndash886 2012
Mathematical Problems in Engineering 5
[11] D Ryan and F B Hanson ldquoOptimal harvesting of a logisticpopulation in an environment with stochastic jumpsrdquo Journalof Mathematical Biology vol 24 no 3 pp 259ndash277 1986
[12] I-S Wee ldquoStability for multidimensional jump-diffusion pro-cessesrdquo Stochastic Processes and Their Applications vol 80 no2 pp 193ndash209 1999
[13] H Kunita ldquoItorsquos stochastic calculus its surprising power forapplicationsrdquo Stochastic Processes and Their Applications vol120 no 5 pp 622ndash652 2010
[14] J Bao X Mao G Yin and C Yuan ldquoCompetitive Lotka-Volterra population dynamics with jumpsrdquo Nonlinear AnalysisTheory Methods amp Applications Series A Theory and Methodsvol 74 no 17 pp 6601ndash6616 2011
[15] L H R Alvarez and L A Shepp ldquoOptimal harvesting ofstochastically fluctuating populationsrdquo Journal of MathematicalBiology vol 37 no 2 pp 155ndash177 1998
[16] W Li and K Wang ldquoOptimal harvesting policy for generalstochastic logistic population modelrdquo Journal of MathematicalAnalysis and Applications vol 368 no 2 pp 420ndash428 2010
[17] W Li K Wang and H Su ldquoOptimal harvesting policy forstochastic Logistic populationmodelrdquoAppliedMathematics andComputation vol 218 no 1 pp 157ndash162 2011
[18] A Dixit The Art of Smooth Pasting Harwood Academic Pub-lishers Chur Switzerland 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
[11] D Ryan and F B Hanson ldquoOptimal harvesting of a logisticpopulation in an environment with stochastic jumpsrdquo Journalof Mathematical Biology vol 24 no 3 pp 259ndash277 1986
[12] I-S Wee ldquoStability for multidimensional jump-diffusion pro-cessesrdquo Stochastic Processes and Their Applications vol 80 no2 pp 193ndash209 1999
[13] H Kunita ldquoItorsquos stochastic calculus its surprising power forapplicationsrdquo Stochastic Processes and Their Applications vol120 no 5 pp 622ndash652 2010
[14] J Bao X Mao G Yin and C Yuan ldquoCompetitive Lotka-Volterra population dynamics with jumpsrdquo Nonlinear AnalysisTheory Methods amp Applications Series A Theory and Methodsvol 74 no 17 pp 6601ndash6616 2011
[15] L H R Alvarez and L A Shepp ldquoOptimal harvesting ofstochastically fluctuating populationsrdquo Journal of MathematicalBiology vol 37 no 2 pp 155ndash177 1998
[16] W Li and K Wang ldquoOptimal harvesting policy for generalstochastic logistic population modelrdquo Journal of MathematicalAnalysis and Applications vol 368 no 2 pp 420ndash428 2010
[17] W Li K Wang and H Su ldquoOptimal harvesting policy forstochastic Logistic populationmodelrdquoAppliedMathematics andComputation vol 218 no 1 pp 157ndash162 2011
[18] A Dixit The Art of Smooth Pasting Harwood Academic Pub-lishers Chur Switzerland 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![OPTIMAL STOPPING PROBLEMS FOR THE MAXIMUM By Curdin … · general theory of optimal stopping [15] suggests that the optimal stopping time is the first entry time of the process](https://static.fdocuments.in/doc/165x107/5fb27fe98ce835194c58a36a/optimal-stopping-problems-for-the-maximum-by-curdin-general-theory-of-optimal-stopping.jpg)