# An Optimal Stopping Problem for Jump Diffusion Logistic Population Model

Authors: Yang Sun; Xiaohui Ai
Journal: Mathematical Problems in Engineering
 (2016)

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## 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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## Body

## 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(1)dXt=rXt-bXt2dt, 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,(2)dXt=rXt-bXt2dt+μXtdBt,X0=x≥0,t≥0, 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(3)dXt=rXt-bXt2dt+μXtdBt+∫0∞cXt-N~dt,dz,X0=x≥0,t≥0,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 Section2, 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 sizeXt at time t is given by the stochastic logistic population model with Poisson jump(4)dXt=rXt-bXt2dt+μXtdBt+∫0∞cXt-N~dt,dz.It can be proved that ifr>0, b>0 and μ,c are bounded constants, then (4) has a unique positive solution Xt defined by(5)Xtx=Φt1/x+b∫0tΦsds,where (6)Φt=exp⁡r-12μ2t+μBt+∫Yln1+c-ctvdz+∫0t∫Yln1+cN~ds,dzfor 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: findv∗(s,x) and τ∗ such that(7)v∗s,x=supτ⁡Exe-ρτpXτβ-q=Exe-ρτ∗pXτ∗β-q,β,p,q>0; the sup is taken over all stopping times τ of the process Xt, t>0 with the reward function (8)Rs,x=e-ρspxβ-q, where the discounted exponent is ρ>0, e-ρs(pxβ-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 timeτ∗ given in (30) with the optimal stopping boundary x∗ from (23) on the interval (0,K) such that we can obtain the optimal profit v∗ 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(9)dXt=rXt-bXt2dt+μXtdBt+∫0∞cXt-N~dt,dz,and applying the Itô formula to a C2-function f such that E[∫0t∫Yft,zv(dz)dt]<∞ and f′,f′′ are bounded, we have the infinitesimal generator of the process f(Xt), that is(10)Lfx=rx-bx2f′x+12μ2x2f′′x+∫Yfx+cx-fx-cxf′xvdz;provided that ∫Y{f(x+cx)-f(x)-cxf′(x)}v(dz) is well defined since(11)fx+cx-fx-cxf′x≤12c2x2f′′x and cx are bounded.Now let us consider a function equation(12)Lfx=0,x∈R+.We can try a solution of the formf(x)=αxβ, x∈R+ to determine the unknown function; that is(13)Lfx=αxββr-bx+12ββ-1μ2+∫Y1+cβ-1-βcvdz≜fxgβ=0,where (14)gβ=βr-bx+12ββ-1μ2+∫Y1+cβ-1-βcvdzis well defined.Lemma 1.
g ( β ) = 0 has two distinct real roots, the largest one, β2, of which satisfies (15)0<β2<1.Proof.
The functiong(β) is decomposed into the sum of two functions(16)g1β=β12β-1μ2+r-bx,∀x<K,g2β=∫Y1+cβ-1-βcvdz.
Since the formerg1 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(17)g0=0,g1=bK-x>0;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 (18) 0 < β < β 2 .
Assumption 2 (19) 1 2 μ 2 p β β - 1 + ∫ Y p 1 + c β - p - c p β v d z < 0 .
Now, let us define a functionf∗:R+→R by(20)f∗x=fx=α∗xβ2,0<x<x∗Rx=pxβ-q,x≥x∗,where α∗ and x∗>0 are constants which are uniquely determined by the following equations [2, 18]:
Value matching condition:(21)fx∗=Rx∗,f′x∗=R′x∗.
That is,(22)α∗=qββ2-βqβ2pβ2-ββ2/β,x∗=qβ2pβ2-β1/β.Lemma 2.
Under Assumptions 1 and 2, the functionf∗:R+→R satisfies the following properties (1)–(4).
(1) For anyx∈R+,(23)f∗x≥Rx.
(2)f∗(x) is strictly increasing in x.
(3) For anyx∈R+(x≠x∗),(24)Lf∗x≤0.
(4) For anyx∈R+, either ineq. (23) or (24) holds with equality.Proof.
(1) Settingy(x)=f(x)-R(x)=α∗xβ2-pxβ+q and differentiating y(x) with respect to x, we have(25)y′x=f′x-R′x=xβ-1α∗β2xβ2-β-pβ<0,0<x<x∗≥0,x≥x∗;hence, under Assumption 1, α∗β2xβ2-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∗)=0,y(+∞)=+∞.
(2) It is obvious.
(3) For0<x<x∗, f∗(x)=f(x), we have (26)Lf∗x=Lfx=0from (12).
Forx>x∗, f∗(x)=R(x)=pxβ-q, we obtain(27)Lfx=LRx=xβpβr-bx+12μ2pββ-1+∫Yp1+cβ-p-cpβvdz=pβxβr-bx+xβ12μ2pββ-1+∫Yp1+cβ-p-cpβvdz≤0,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 functionf∗:R+→R is the optimal value function; that is,(28)v∗x=f∗x,x∈R+.Moreover, the optimal stopping regionA(⊂R+) and the optimal stopping time τ∗ are given by the following:(29)A=x∈R+:f∗x=Rx=x∗,+∞,(30)τ∗≔inf⁡t∈R+:Xt∗∈A.Proof.
Using the functionf∗:R+→R, we define a new stochastic process M={Mt;t∈R+} by(31)Mt=e-αtf∗Xtx-f∗x+∫0te-αsLf∗Xsxds+Mt~,where(32)Mt~=∫0tμXtxf∗′XtxdBt+∫0t∫Yf∗Xtx+cXtx-f∗XtxN~dt,dzis a continuous local martingale, and by applying the Itô formula for the process e-αtf∗(Xtx), we obtain Mt=0.
Lemma2 (4) implies(33)e-αtf∗Xtx≤f∗x+M~t,with the help of the optimal sample theorem for martingale; we have, for any stopping time τ for the process {M~t}t≥0,(34)Ee-αt∧τf∗Xt∧τx≤f∗x+EM~t∧τ,which can be written by(35)Ee-αt∧τf∗Xt∧τx≤f∗x,by noting the obvious fact EM~t∧τ=0.
Takinglim⁡inft→∞ of both sides of (35), we have, by Fatou lemma,(36)Ee-ατf∗XtxIτ<∞≤f∗x;moreover, since the function f∗ has property Lemma 2 (1), it holds that(37)Ee-ατRXtxIτ<∞≤Ee-ατf∗XtxIτ<∞≤f∗x.
On the other hand, for the stopping timeτ∗ defined by (30)(38)Ee-ατ∗∧tf∗Xτ∗∧tx=f∗x.
By the properties of Lemma2 (1)–(4) of the function f∗, we assure that A=[x∗,∞) and by the properties of Lemma 2 (2), it holds that(39)0<f∗Xτ∗∧tx≤f∗x∗,a.s.
Takinglimt∈∞ of both sides of (38), we have, by the bounded convergence theorem of Lebesgue,(40)f∗x=Ee-ατ∗f∗Xτ∗xIτ∗<∞=Ee-ατ∗RXτ∗xIτ∗<∞,where the second equality follows from the fact that, on the event {τ∗<∞},(41)f∗Xτ∗x=RXτ∗x.
Then we conclude that(42)f∗≤Ee-ατ∗RXτ∗xIτ∗<∞≤f∗;that is,(43)v∗x=f∗x=Ee-ατ∗RXτ∗xIτ∗<∞.



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*Source: 5839672-2016-08-09.xml*