An Optimal Stopping Problem for Jump Diffusion Logistic Population Model

This paper examines an optimal stopping problem for the stochastic (Wiener-Poisson) jumpdiffusion logistic populationmodel.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.


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 where   denotes the density of resource population at time ,  > 0 is called the intrinsic growth rate, and  = / > 0 ( 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, 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.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 where ( − ) is the left limit of (), , , , and   are defined in (2),  is a bounded constant,  is a Poisson counting measure with characteristic measure V on a measurable subset  of (0, ∞) with V() < ∞, and Ñ(, ) = (, ) − V().Throughout the paper, we assume that  and  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   at time  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.

Description of Problem
Suppose the population with size   at time  is given by the stochastic logistic population model with Poisson jump It can be proved that if  > 0,  > 0 and ,  are bounded constants, then (4) has a unique positive solution   defined by where for all  ≥ 0 (see Bao et al. [14]) and note that 0 ≤   < .
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 * (, ) and  * such that the sup is taken over all stopping times  of the process   ,  > 0 with the reward function where the discounted exponent is  > 0,  − (  − ) is the profit of selling fish at time , and  represents a fixed fee and it is nature to assume that  < .  denotes the expectation with respect to the probability law   of the process   ,  ≥ 0 starting at  0 =  > 0.
We will search for an optimal stopping time  * given in (30) with the optimal stopping boundary  * from (23) on the interval (0, ) such that we can obtain the optimal profit V * in (28) and the optimal stopping region  in (29).Note that it is trivial that the initial value  ≤ , so we further assume  > .

Analysis
For the jump diffusion logistic population model and applying the Itô formula to a  2 −function  such that E[∫  0 ∫  |(, )|V()] < ∞ and   ,   are bounded, we have the infinitesimal generator of the process (  ), that is and  are bounded.Now let us consider a function equation We can try a solution of the form () =   ,  ∈ R + to determine the unknown function; that is where is well defined.
Lemma 1. () = 0 has two distinct real roots, the largest one,  2 , of which satisfies Proof.The function () is decomposed into the sum of two functions Since the former  1 is a mixture of convex exponential function (1+)  ( is bounded), we assume that () is strictly convex function.Furthermore, we have therefore, the nonlinear equation () = 0 has two distinct real roots  1 ,  2 such that  1 = 0 and 0 <  2 < 1, respectively.We assume the following.
Assumption 2 Now, let us define a function  * : R + → R by where  * and  * > 0 are constants which are uniquely determined by the following equations [2,18]: Value matching condition: That is, Lemma 2. Under Assumptions 1 and 2, the function  * : R + → R satisfies the following properties ( 1)-( 4).