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.
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 where denotes the density of resource population at time , is called the intrinsic growth rate, and ( 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 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 jumpwhere is the left limit of , , , , and are defined in (2), is a bounded constant, is a Poisson counting measure with characteristic measure on a measurable subset of with , and . 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.
2. 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 , and are bounded constants, then (4) has a unique positive solution defined bywhere for all (see Bao et al. [14]) and note that .
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 and such that the sup is taken over all stopping times of the process , with the reward function where the discounted exponent is , 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 , starting at
We will search for an optimal stopping time given in (30) with the optimal stopping boundary from (23) on the interval such that we can obtain the optimal profit in (28) and the optimal stopping region in (29). Note that it is trivial that the initial value , so we further assume
3. Analysis
For the jump diffusion logistic population modeland applying the Itô formula to a function such that and are bounded, we have the infinitesimal generator of the process , that isprovided that is well defined since and are bounded.
Now let us consider a function equation
We can try a solution of the form , to determine the unknown function; that iswhere is well defined.
Lemma 1. has two distinct real roots, the largest one, , of which satisfies
Proof. The function is decomposed into the sum of two functionsSince the former is a mixture of convex exponential function ( is bounded), we assume that is strictly convex function. Furthermore, we havetherefore, the nonlinear equation has two distinct real roots such that and , respectively.
We assume the following.
Assumption 1Assumption 2Now, let us define a function bywhere and 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 satisfies the following properties (1)–(4).
(1) For any ,(2) is strictly increasing in
(3) For any ,(4) For any , either ineq. (23) or (24) holds with equality.
Proof. (1) Setting and differentiating with respect to , we havehence, under Assumption 1, is an increasing function on , and we obtain the conclusion with the help of the fact that .
(2) It is obvious.
(3) For , , we have from (12).
For , , we obtainunder Assumption 2. We finished the proof of ().
(4) It is trivial from () and ().
Now let us give the main theorem.
Theorem 3. Under Assumptions 1 and 2, the function is the optimal value function; that is,
Moreover, the optimal stopping region and the optimal stopping time are given by the following:
Proof. Using the function , we define a new stochastic process bywhereis a continuous local martingale, and by applying the Itô formula for the process , we obtain
Lemma 2 () implieswith the help of the optimal sample theorem for martingale; we have, for any stopping time for the process ,which can be written byby noting the obvious fact
Taking of both sides of (35), we have, by Fatou lemma,moreover, since the function has property Lemma 2 (), it holds thatOn the other hand, for the stopping time defined by (30) By the properties of Lemma 2 ()–() of the function , we assure that and by the properties of Lemma 2 (), it holds thatTaking of both sides of (38), we have, by the bounded convergence theorem of Lebesgue,where the second equality follows from the fact that, on the event ,Then we conclude thatthat is,
Competing Interests
The authors declare that they have no competing interests.
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).