An Optimal Stopping Problem for Jump Diffusion Logistic Population Model

Abstract

This 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.

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Yang Sun 1 and Xiaohui Ai 2

Academic Editor:Khaled Bahlali

1, School of Applied Science, Harbin University of Science and Technology, Harbin 150080, China

2, Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received 11 May 2016; Revised 13 July 2016; Accepted 21 July 2016

This 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.

1. Introduction

The 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 incorporating random fluctuations in either population size or model parameters have been the subject of much interest. The 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 [figure omitted; refer to PDF] where _Xt denotes the density of resource population at time t, r>0 is called the intrinsic growth rate, and b=r/K>0 (K is the environmental carrying capacity). The 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, [figure omitted; refer to PDF] where the constants r, b are mentioned in (1), μ is a measure of the size of the noise in the system, and _Bt is 1-dimensional Brownian motion defined on a complete probability space (Ω,F,_{{Ft }t≥0} ,P) satisfing the usual conditions. There 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 environmental 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 [figure omitted; refer to PDF] where X(^t- ) is the left limit of X(t), r, b, μ, and _Bt are defined in (2), c is a bounded constant, N is a Poisson counting measure with characteristic measure v on a measurable subset Y of (0,∞) with v(Y)<∞, and N~(dt,dz)=N(dt,dz)-v(dz)dt. Throughout the paper, we assume that B and N are independent. More discussions of the stochastic jump diffusion model 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, time averaging methods, and stochastic calculus are used on optimal harvesting problems for model (2); all the aforementioned works can be found in Alvarez and Shepp [15], Li and Wang [16], and Li et al. [17]. To my best knowledge, even for model (2), there is little try by using optimal stopping theory on optimal harvesting problems; therefore, in this paper, we will try the optimal stopping approach to solve the optimal harvesting problem for model (3), which is the motivation of the paper.

The paper is organized as follows. In Section 2, in order to find the optimal value function and the optimal stopping region, we formulate the problem and suppose we have a fish factory with a population (e.g., a fish population in a pond) whose size _Xt at time t is described by the stochastic jump diffusion model (3), as a stopping problem. In Section 3, an explicit function for the value function is verified; meanwhile, the optimal stopping time and the optimal stopping region are expressed.

2. Description of Problem

Suppose the population with size _Xt at time t is given by the stochastic logistic population model with Poisson jump [figure omitted; refer to PDF]

It can be proved that if r>0, b>0 and μ,c are bounded constants, then (4) has a unique positive solution _Xt defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for all t≥0 (see Bao et al. [14]) and note that 0<=_Xt <K.

Supposing that the population is, say, a fish population in a pond, the goal of this paper, the optimal strategy for selling a fish factory, can be considered as an optimal stopping problem: find ^{v[low *]} (s,x) and ^{τ[low *]} such that [figure omitted; refer to PDF] the sup is taken over all stopping times τ of the process _Xt , t>0 with the reward function [figure omitted; refer to PDF] where the discounted exponent is ρ>0, ^e-ρs (p^xβ -q) is the profit of selling fish at time τ, and q represents a fixed fee and it is nature to assume that q<K. ^Ex denotes the expectation with respect to the probability law ^Px of the process _Xt , t≥0 starting at _X0 =x>0.

We will search for an optimal stopping time ^{τ[low *]} given in (30) with the optimal stopping boundary ^{x[low *]} from (23) on the interval (0,K) such that we can obtain the optimal profit ^{v[low *]} in (28) and the optimal stopping region A in (29). Note that it is trivial that the initial value x<=q, so we further assume x>q.

3. Analysis

For the jump diffusion logistic population model [figure omitted; refer to PDF] and applying the Itô formula to a ^C2 -function f such that E[^∫0t∫Y (f(t,z))v(dz)dt]<∞ and ^{f[variant prime]} ,^{f[variant prime][variant prime]} are bounded, we have the infinitesimal generator of the process f(_Xt ), that is [figure omitted; refer to PDF] provided that ^∫Y {f(x+cx)-f(x)-cx^{f[variant prime]} (x)}v(dz) is well defined since [figure omitted; refer to PDF] and cx are bounded.

Now let us consider a function equation [figure omitted; refer to PDF]

We can try a solution of the form f(x)=α^xβ , x∈_R+ to determine the unknown function; that is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is well defined.

Lemma 1.

g ( β ) = 0 has two distinct real roots, the largest one, _β2 , of which satisfies [figure omitted; refer to PDF]

Proof.

The function g(β) is decomposed into the sum of two functions [figure omitted; refer to PDF]

Since the former _g1 is a mixture of convex exponential function (1+c^)β (c is bounded), we assume that g(β) is strictly convex function. Furthermore, we have [figure omitted; refer to PDF] therefore, the nonlinear equation g(β)=0 has two distinct real roots _β1 ,_β2 such that _β1 =0 and 0<_β2 <1, respectively.

We assume the following.

Assumption 1 [figure omitted; refer to PDF]

Assumption 2 [figure omitted; refer to PDF]

Now, let us define a function ^{f[low *]} :_R+ [arrow right]R by [figure omitted; refer to PDF] where ^{α[low *]} and ^{x[low *]} >0 are constants which are uniquely determined by the following equations [2, 18]:

Value matching condition: [figure omitted; refer to PDF]

That is, [figure omitted; refer to PDF]

Lemma 2.

Under Assumptions 1 and 2, the function ^{f[low *]} :_R+ [arrow right]R satisfies the following properties (1 )-(4 ).

(1) For any x∈_R+ , [figure omitted; refer to PDF]

(2) ^{f[low *]} (x) is strictly increasing in x.

(3) For any x∈_R+ (x≠^{x[low *]} ), [figure omitted; refer to PDF]

(4) For any x∈_R+ , either ineq. (23) or (24) holds with equality.

Proof.

(1) Setting y(x)=f(x)-R(x)=^{α[low *]x_β2} -p^xβ +q and differentiating y(x) with respect to x, we have [figure omitted; refer to PDF] hence, under Assumption 1, ^{α[low *]}_β2^{x_β2 -β} -pβ is an increasing function on [0,∞), and we obtain the conclusion with the help of the fact that y(0)=q,y(^{x[low *]} )=0,y(+∞)=+∞.

(2) It is obvious.

(3) For 0<x<^{x[low *]} , ^{f[low *]} (x)=f(x), we have [figure omitted; refer to PDF] from (12).

For x>^{x[low *]} , ^{f[low *]} (x)=R(x)=p^xβ -q, we obtain [figure omitted; refer to PDF] 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 ^{f[low *]} :_R+ [arrow right]R is the optimal value function; that is, [figure omitted; refer to PDF]

Moreover, the optimal stopping region A(⊂_R+ ) and the optimal stopping time ^{τ[low *]} are given by the following: [figure omitted; refer to PDF]

Proof.

Using the function ^{f[low *]} :_R+ [arrow right]R, we define a new stochastic process M={_Mt ;t∈_R+ } by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a continuous local martingale, and by applying the Itô formula for the process ^{e-αtf[low *]} (^Xtx ), we obtain _Mt =0.

Lemma 2 (4) implies [figure omitted; refer to PDF] with the help of the optimal sample theorem for martingale; we have, for any stopping time τ for the process _{{M~t }t≥0} , [figure omitted; refer to PDF] which can be written by [figure omitted; refer to PDF] by noting the obvious fact E_M~t⋀τ =0.

Taking _{lim[...]inft[arrow right]∞} of both sides of (35), we have, by Fatou lemma, [figure omitted; refer to PDF] moreover, since the function ^{f[low *]} has property Lemma 2 (1), it holds that [figure omitted; refer to PDF]

On the other hand, for the stopping time ^{τ[low *]} defined by (30) [figure omitted; refer to PDF]

By the properties of Lemma 2 (1)-(4) of the function ^{f[low *]} , we assure that A=[^{x[low *]} ,∞) and by the properties of Lemma 2 (2), it holds that [figure omitted; refer to PDF]

Taking _limt∈∞ of both sides of (38), we have, by the bounded convergence theorem of Lebesgue, [figure omitted; refer to PDF] where the second equality follows from the fact that, on the event {^{τ[low *]} <∞}, [figure omitted; refer to PDF]

Then we conclude that [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF]

Acknowledgments

Yang Sun is supported by the NSFC Grant (no. 51406044 and no. 11401085) and Natural Science Foundation of the Education Department of Heilongjiang Province (Grant no. 12521116). Xiaohui Ai is supported by the NSFC Grant (no. 11401085) and the Fundamental Research Funds for the Central Universities (no. 2572015BB14).

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