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An Optimal Stopping Problem for Jump Diffusion Logistic Population Model.

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

d[X.sub.t] = (r[X.sub.t] - b[X.sup.2.sub.t])dt, (1)

where [X.sub.t] 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,

[mathematical expression not reproducible], (2)

where the constants r, b are mentioned in (1), [mu] is a measure of the size of the noise in the system, and [B.sub.t] is 1-dimensional Brownian motion defined on a complete probability space ([OMEGA], [??], [{[[??].sub.t]}.sub.t[greater than or equal to]0], P) satisfing the usual conditions. There are so many extensive researches in literature, such as Lungu and Oksendal [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

[mathematical expression not reproducible], (3)

where X([t.sup.-]) is the left limit of X(t), r, b, y,, and [B.sub.t] 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, [infinity]) with v(Y) < [infinity], and [??](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 [X.sub.t] 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 [X.sub.t] at time t is given by the stochastic logistic population model with Poisson jump

[mathematical expression not reproducible]. (4)

It can be proved that if r > 0, b > 0 and [mu], c are bounded constants, then (4) has a unique positive solution [X.sub.t] defined by

[mathematical expression not reproducible], (5)

where

[mathematical expression not reproducible] (6)

for all t [greater than or equal to] 0 (see Bao et al. [14]) and note that 0 [less than or equal to] [X.sub.t] < 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.sup.*](s, x) and [[tau].sup.*] such that

[mathematical expression not reproducible]; (7)

the sup is taken over all stopping times t of the process [X.sub.t], t > 0 with the reward function

R(s, x) = [e.sup.-[rho]s] (p[x.sup.[beta]] - q), (8)

where the discounted exponent is [rho] > 0, [e.sup.-[rho]s](p[x.sup.[beta]] - q) is the profit of selling fish at time [tau], and q represents a fixed fee and it is nature to assume that q < K. [E.sup.x] denotes the expectation with respect to the probability law [P.sup.x] of the process [X.sub.t], t [greater than or equal to] 0 starting at [X.sub.0] = x > 0.

We will search for an optimal stopping time [[tau].sup.*] given in (30) with the optimal stopping boundary [x.sup.*] from (23) on the interval (0, K) such that we can obtain the optimal profit [v.sup.*] in (28) and the optimal stopping region A in (29). Note that it is trivial that the initial value x [less than or equal to] q, so we further assume x > q.

3. Analysis

For the jump diffusion logistic population model

[mathematical expression not reproducible], (9)

and applying the Ito formula to a [C.sup.2]-function f such that E[[[integral].sup.t.sub.0][[integral].sub.Y][absolute value of f(t, z)]v(dz)dt] < [infinity] and f', f" are bounded, we have the infinitesimal generator of the process f([X.sub.t]), that is

[mathematical expression not reproducible]; (10)

provided that [[integral].sub.Y]{f(x + cx) - f(x) - cxf'(x)}v(dz) is well defined since

[mathematical expression not reproducible] (11)

and cx are bounded.

Now let us consider a function equation

[Lf](x) = 0, x [member of] [R.sub.+]. (12)

We can try a solution of the form f(x) = [alpha][x.sup.[beta]], x [member of] [R.sub.+] to determine the unknown function; that is

[mathematical expression not reproducible], (13)

where

[mathematical expression not reproducible] (14)

is well defined.

Lemma 1. g([beta]) = 0 has two distinct real roots, the largest one, [[beta].sub.2], of which satisfies

0 < [[beta].sub.2] < 1. (15)

Proof. The function g([beta]) is decomposed into the sum of two functions

[mathematical expression not reproducible]. (16)

Since the former [g.sub.1] is a mixture of convex exponential function [(1+c).sup.[beta]] (c is bounded), we assume that g([beta]) is strictly convex function. Furthermore, we have

g(0) = 0, g(1) = b(K - x) > 0; (17)

therefore, the nonlinear equation g([beta]) = 0 has two distinct real roots [[beta].sub.1], [[beta].sub.2] such that [[beta].sub.1] = 0 and 0 < [[beta].sub.2] < 1, respectively.

We assume the following.

Assumption 1

0 < [beta] < [[beta].sub.2]. (18)

Assumption 2

[mathematical expression not reproducible]. (19)

Now, let us define a function [f.sup.*] : [R.sub.+] [right arrow] R by

[mathematical expression not reproducible], (20)

where [[alpha].sup.*] and [x.sup.*] > 0 are constants which are uniquely determined by the following equations [2, 18]:

Value matching condition:

f([x.sup.*]) = R([x.sup.*]), f'([x.sup.*]) = R'([x.sup.*]). (21)

That is,

[mathematical expression not reproducible]. (22)

Lemma 2. Under Assumptions 1 and 2, the function [f.sup.*] : [R.sub.+] [right arrow] R satisfies the following properties (1)-(4).

(1) For any x [member of] [R.sub.+],

[f.sup.*](x) [greater than or equal to] R(x). (23)

(2) [f.sup.*](x) is strictly increasing in x.

(3) For any x [member of] [R.sub.+] (x [not equal to] [x.sup.*]),

[L[f.sup.*]](x) [less than or equal to] 0. (24)

(4) For any x [member of] [R.sub.+], either ineq. (23) or (24) holds with equality.

Proof.

(1) Setting [mathematical expression not reproducible] and differentiating y(x) with respect to x, we have

[mathematical expression not reproducible]; (25)

hence, under Assumption 1, [mathematical expression not reproducible] is an increasing function on [0, [infinity]), and we obtain the conclusion with the help of the fact that y(0) = q, y([x.sup.*]) = 0, y(+[infinity]) = +[infinity].

(2) It is obvious.

(3) For 0 < x < [x.sup.*], [f.sup.*](x) = f(x), we have

[L[f.sup.*]](x) = [Lf](x) = 0 (26)

from (12).

For x > [x.sup.*], [f.sup.*](x) = R(x) = p[x.sup.[beta]] - q, we obtain

[mathematical expression not reproducible], (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 [f.sup.*] : [R.sub.+] [right arrow] R is the optimal value function; that is,

[v.sup.*](x) = [f.sup.*](x), x [member of] [R.sub.+]. (28)

Moreover, the optimal stopping region A([subset] [R.sub.+]) and the optimal stopping time [[tau].sup.*] are given by the following:

A = {x [member of] [R.sub.+] : [f.sup.*](x) = R(x)} = {[x.sup.*], +[infinity]}, (29)

[[tau].sup.*] := inf [t [member of] [R.sub.+] : [X.sup.*.sub.t] [member of] A}. (30)

Proof. Using the function [f.sup.*] : [R.sub.+] [right arrow] R, we define a new stochastic process M = {[M.sub.t]; t [member of] [R.sub.+]} by

[mathematical expression not reproducible], (31)

where

[mathematical expression not reproducible] (32)

is a continuous local martingale, and by applying the Ito formula for the process [e.sup.-[alpha]t][f.sup.*]([X.sup.x.sub.t]), we obtain [M.sub.t] = 0.

Lemma 2 (4) implies

[e.sup.-[alpha]t][f.sup.*]([X.sup.x.sub.t]) [less than or equal to] [f.sup.*](x) + [[??].sub.t], (33)

with the help of the optimal sample theorem for martingale; we have, for any stopping time t for the process [{[[??].sub.t]}.sub.t[greater than or equal to]0],

[mathematical expression not reproducible], (34)

which can be written by

[mathematical expression not reproducible], (35)

by noting the obvious fact E[[??].sub.t[conjunction][tau]] = 0.

Taking lim [inf.sub.t[right arrow][infinity]] of both sides of (35), we have, by Fatou lemma,

[mathematical expression not reproducible]; (36)

moreover, since the function [f.sup.*] has property Lemma 2 (1), it holds that

[mathematical expression not reproducible]. (37)

On the other hand, for the stopping time [[tau].sup.*] defined by (30)

[mathematical expression not reproducible]. (38)

By the properties of Lemma 2 (1)-(4) of the function [f.sup.*], we assure that A = [[x.sup.*], [infinity]) and by the properties of Lemma 2 (2), it holds that

[mathematical expression not reproducible]. (39)

Taking [lim.sub.t[member of][infinity]] of both sides of (38), we have, by the bounded convergence theorem of Lebesgue,

[mathematical expression not reproducible], (40)

where the second equality follows from the fact that, on the event {[[tau].sup.*] < [infinity]},

[mathematical expression not reproducible]. (41)

Then we conclude that

[mathematical expression not reproducible]; (42)

that is,

[mathematical expression not reproducible]. (43)

http://dx.doi.org/10.1155/2016/5839672

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).

References

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[10] M. Liu and K. Wang, "On a stochastic logistic equation with impulsive perturbations," Computers & Mathematics with Applications, vol. 63, no. 5, pp. 871-886, 2012.

[11] D. Ryan and F. B. Hanson, "Optimal harvesting of a logistic population in an environment with stochastic jumps," Journal of Mathematical Biology, vol. 24, no. 3, pp. 259-277, 1986.

[12] I.-S. Wee, "Stability for multidimensional jump-diffusion processes," Stochastic Processes and Their Applications, vol. 80, no. 2, pp. 193-209, 1999.

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[14] J. Bao, X. Mao, G. Yin, and C. Yuan, "Competitive Lotka-Volterra population dynamics with jumps," Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods, vol. 74, no. 17, pp. 6601-6616, 2011.

[15] L. H. R. Alvarez and L. A. Shepp, "Optimal harvesting of stochastically fluctuating populations," Journal of Mathematical Biology, vol. 37, no. 2, pp. 155-177, 1998.

[16] W. Li and K. Wang, "Optimal harvesting policy for general stochastic logistic population model," Journal of Mathematical Analysis and Applications, vol. 368, no. 2, pp. 420-428, 2010.

[17] W. Li, K. Wang, and H. Su, "Optimal harvesting policy for stochastic Logistic population model," Applied Mathematics and Computation, vol. 218, no. 1, pp. 157-162, 2011.

[18] A. Dixit, The Art of Smooth Pasting, Harwood Academic Publishers, Chur, Switzerland, 1993.

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; sunysy@126.com

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

Academic Editor: Khaled Bahlali
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Title Annotation:Research Article
Author:Sun, Yang; Ai, Xiaohui
Publication:Mathematical Problems in Engineering
Date:Jan 1, 2016
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