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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 831082, 13 pages
Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps
1School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410075, China
2Department of Mathematics, Huaihua University, Huaihua, Hunan 418008, China
Received 19 November 2011; Accepted 30 November 2011
Academic Editor: Shaher Momani
Copyright © 2012 Qiyong Li and Siqing Gan. 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.
This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize when , and they are exponentially mean-square stable if the stepsize when . Finally, some numerical experiments are given to illustrate the theoretical results.
Models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering. In particular, they are used in mathematical finance in order to simulate asset prices, interest rates, and volatilities [1, 2]. Jump models also arise in many other application areas and have proved successful at describing unexpected, abrupt changes of state . So, it is valuable to investigate the properties of the solutions of these problems.
As is well known, explicit solutions of stochastic differential equations can rarely be obtained. It is necessary to construct efficient numerical methods to solve these equations. In recent years, many researchers worked on the construction of numerical schemes for stochastic ordinary differential equations (SODEs) (see [4, 5], and their references) and stochastic delay differential equations (SDDEs), see, for example, [6–11] and references therein. For SODEs with jumps, the strong convergence and mean-square stability of some semi-implicit numerical methods are investigated in [12–15]. A compensated split-step backward Euler method for SODEs with jumps is introduced in  and proved to satisfy a better stability property than the split-step backward Euler method.
For SDDEs with jumps, most of the existing work is concerned about convergence property of numerical methods, see, for example, [16–19]. There are few results on stability property, which motivates our work. In , Tan and Wang investigated the mean-square stability of the explicit Euler method for linear SDDEs with jumps. The aim of our paper is to investigate the mean-square stability of the compensated stochastic methods for nonlinear SDDEs with jumps.
This paper is organized as follows. In Section 2, we obtain a stability result for the analytical solution of (2.1). In Section 3, the compensated stochastic methods are constructed to solve problem (2.1). In Section 4, our main results will be stated and proved. It is shown that the compensated stochastic methods inherit mean-square stability of the exact solution. More precisely, the methods are mean-square stable for any stepsize when , and they are exponentially mean-square stable if the stepsize when . Moreover, when , the method is exponentially mean-square stable for every stepsize . Finally, some numerical experiments are reported to illustrate the theoretical results.
2. Stability of the Analytical Solution
Throughout this paper, we let ([−,0]; ) denote the family of continuous functions from to equipped with the norm , where is the Euclidean norm in . Denoted by the family of all bounded, measurable, ([,0]; ) valued stochastic variables. The inner product of in is denoted by . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by .
We consider the nonlinear SDDEs with jumps in Itô’s sense of the form: with initial data and is a constant, denotes , , , are continuous functions, and . is an -dimensional Wiener process defined on the complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). is a scalar Poisson process with parameter defined on the same probability space. Assume that and are independent of . Moreover, we assume that is -measurable and right continuous with . We also assume that and , so problem (2.1) admits a zero solution .
Definition 2.1 (see ). The zero solution of (2.1) is said to be pth moment exponentially stable if there is a pair of positive constants and such that for all . When , it is usually said to be exponentially mean-square stable.
Now, we establish a mean-square stability condition for problem (2.1).
Theorem 2.2. Suppose that there are some constants , such that for all . If then the zero solution of (2.1) is exponentially mean-square stable.
Proof. Let , , it follows from Itô’s formula that where . Taking expectation and using the properties of Itô integral give From (2.3) and (2.4), we have Similarly, (2.5) and (2.6) yield Substituting (2.10)-(2.11) into (2.9) yields Let , , we have where Moreover, implies . By Lemma 1.1 in , there exist positive constants and such that Hence, the theorem is proven.
Based on the above result, we are going to study the stability of numerical methods for (2.1) in the following sections.
3. Compensated Stochastic Methods for Nonlinear SDDEs with Jumps
Since the compensated Poisson process is a martingale satisfying the property we rewrite problem (2.1) in an equivalent form: where is defined as Applying the stochastic methods to (3.2) leads to the following compensated stochastic methods: Here, denotes that the approximation to , is a parameter with , is the stepsize which satisfies for a positive integer , and . In particular, Note that for , the numerical solutions in (3.4) are defined by implicit equations. However, due to the one-sided Lipschitz condition (2.3), (3.4) has a unique solution, with probability one, for all , see, for example, [23, Theorem 14.2] and (19) in .
Remark 3.1. Since implies , then the compensated stochastic methods (3.4) produce a well-defined, unique solution if the stability condition holds.
Definition 3.2. For a give stepsize , a numerical method on the nonlinear SDDEs with jumps (2.1) is said to be exponentially mean-square stable, if there exist positive constants and , such that the numerical solution produced by this method satisfies for all initial data .
Definition 3.3. For a give stepsize , a numerical method on the nonlinear SDDEs with jumps (2.1) is said to be mean-square stable if the numerical solution produced by this method satisfies
4. Stability Analysis of the Numerical Solutions
In this section, we study mean-square stability and exponentially mean-square stability of the compensated stochastic methods (3.4). Now, we present the main results of the paper.
Proof. It follows from (3.4) that where Thus, for , we have It follows from (2.3), (2.4), and (2.6) that Note that , and . Furthermore, and are all -measurable. Therefore, we can easily obtain Taking expectation on both sides of (4.3) and substituting (4.4)–(4.7) into (4.3), we have Consequently, by the recursion of inequality (4.8), we have Noting that and , we derive from (4.9) that Rearranging (4.10) and using the notation in (2.7), we obtain Since and , we then derive that the series is convergent, which implies . Consequently, for , the compensated stochastic methods are mean-square stable for any stepsize .
In order to investigate the exponential stability of the numerical methods, we need the following lemma which is Theorem 1 in .
Lemma 4.2 (see ). Suppose, for some fixed integer , that for some and is a sequence of positive numbers that satisfies with if , where . If then , where is a constant.
Now, we present the result as follows.
Theorem 4.3. Suppose that (2.3)–(2.7) hold and the drift coefficient satisfies the linear growth condition, that is, there is a constant such that Define and , where . If , and the stepsize with , then the compensated stochastic methods are exponentially mean-square stable.
Proof. We derive from (3.4) that Hence, we have where is defined as (4.2). From (2.6), (3.3), and (4.14), we obtain Substituting (4.4)–(4.7) and (4.17) into (4.16), and taking expectation, we have which yields Hence, where By Lemma 4.2, we derive that the methods are exponentially mean-square stable if That is, where . Since , there must exist such that when . On the other hand, if is always true, we then define as . Therefore, let , , and , then (4.22) holds when , which completes the proof of Theorem 4.3.
By the proof of Theorem 4.3, we can easily obtain the following result.
5. Numerical Examples
The purpose of this section is to illustrate our theoretical results presented in the previous section by numerical experiments. We first consider the following nonlinear scalar SDDEs with jumps: where is a scalar Poisson process with parameter . In this case, (2.3)–(2.6) are satisfied with , and . So we have in (2.7), which guarantees mean-square stability of the zero solution of (5.1) by Theorem 2.2.
The following numerical experiments will show how the parameter and the stepsize influence the mean-square stability of the compensated stochastic methods. We simulate the expectation of by using 1000 trajectories, that is,
Theorem 4.1 shows that the compensated stochastic methods are mean-square stable for every stepsize when . In Figure 1, we use (3.4) to solve (5.1) and choose the parameter with different values 0.5 and 0.8, and we take the stepsize , and 1, respectively. We can find that the compensated stochastic methods are mean-square stable with these stepsizes.
Theorem 4.3 shows that the compensated stochastic methods are exponentially mean-square stable if the stepsize when . Now, we consider the following nonlinear scalar SDDEs with jumps: where is a scalar Poisson process with parameter . (2.3)–(2.7) and (4.14) are satisfied with , and . Therefore, the zero solution of (5.3) is exponentially mean-square stable. By Theorem 4.3, we calculate , , and . It is easy to see that for every , then we get . Therefore, we obtain , which implies that the methods applied to (5.3) have less restrictions on the stepsize as the value of increases. Now, we use (3.4) to solve (5.3) and choose the parameter and , and we take the stepsize , and 1, respectively. By Theorem 4.3, we compute that when and when . Figure 2 indicates that both methods are exponentially mean-square stable if the stepsize , which is well selected in . It also shows that the methods maybe stable when the stepsize is bigger than , since both methods are stable when stepsize , but they are not stable for stepsize . This indicates that the restriction of the stepsize in Theorem 4.3 is not theoretical optimal. From Figure 2, we also can find that the methods behave better stability when the value of increases and the stepsize decreases.
The authors thank the anonymous referee and the editors for their detailed comments and helpful suggestions. This work was supported by NSF of China (no.11171352) and Hunan Provincial Innovation Foundation for Postgraduates (no. CX2010B118).
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