Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 976183, 11 pages

http://dx.doi.org/10.1155/2014/976183

Research Article

Numerical Solutions to Neutral Stochastic Delay Differential Equations with Poisson Jumps under Local Lipschitz Condition

Jianguo Tan,¹ Hongli Wang,² Yongfeng Guo,¹ and Zhiwen Zhu²

¹Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

²Department of Mechanics, Tianjin University, Tianjin 300072, China

Received 13 February 2014; Revised 18 May 2014; Accepted 19 May 2014; Published 4 June 2014

Academic Editor: Mohamed Abd El Aziz

Copyright © 2014 Jianguo Tan et al. 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.

Abstract.

Recently, Liu et al. (2011) studied the stability of a class of neutral stochastic delay differential equations with Poisson jumps (NSDDEwPJs) by fixed points theory. To the best of our knowledge to date, there are not any numerical methods that have been established for NSDDEwPJs yet. In this paper, we will develop the Euler-Maruyama method for NSDDEwPJs, and the main aim is to prove the convergence of the numerical method. It is proved that the proposed method is convergent with strong order 1/2 under the local Lipschitz condition. Finally, some numerical examples are simulated to verify the results obtained from theory.

1. Introduction

Neutral stochastic delay differential equations (NSDDEs) have recently been studied intensively by, for instance, Kolmanovskii et al. [1, 2], Mao et al. [3–6], Luo et al. [7], Zhou and Hu [8], and Luo [9]. However, explicit solutions can hardly be obtained for NSDDEs; as a result, several numerical schemes have been developed to produce approximate solutions for NSDDEs. For example, Wu and Mao [10] studied the numerical solutions of NSDDEs. Zhang and Gan [11] considered the mean square convergence of one-step methods for NSDDEs. Zhou and Wu [12] studied the convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching. Poisson jumps are becoming increasingly used to model real-world phenomena in different fields such as economics, finance, biology, and physics. There is an extensive literature concerned with Poisson jumps. For example, Wang et al. [13, 14] studied the semi-implicit Euler method for stochastic differential delay equation with jumps (SDDEwJs) and the convergence of numerical solutions to stochastic differential delay equations with Poisson jump and Markovian switching (SDDEwPJMSs). Li et al. [15, 16] discussed the convergence of the numerical solutions for SDDEwJs and SDDEwPJMSs. Luo [17] considered the comparison principle and stability of SDDEwPJMSs. Therefore it is natural and necessary to incorporate jumps in the neutral stochastic delay differential equations. However, the study of neutral stochastic delay differential equations with Poisson jumps is less by far. Cen and Zhou [18] investigated convergence of numerical solutions to neutral stochastic delay differential equation with Poisson jump and Markovian switching. Liu et al. [19] studied the stability of NSDDEwPJs by using fixed points theory. Luo and Taniguchi [20] proved the existence and uniqueness for non-lipschitz stochastic neutral delay evolution equations driven by Poisson jumps. However, there are not any numerical methods that have been established for NSDDEwPJs yet. Therefore, in this paper, we first prove the Euler-Maruyama method applied to NSDDEwPJs converges to the true solution under local Lipschitz condition.

The outline of the paper is as follows. In Section 2 we will introduce some necessary notations and assumptions, and then the Euler-Maruyama method is used to define the numerical solutions for NSDDEwPJs. Section 3 will present several useful lemmas. In Section 4, we state our main result; that is, the numerical solutions will converge to the true solutions of NSDDEwPJs under the local Lipschitz condition. At last, some numerical examples are given to verify the results obtained from the theory.

2. Preliminaries and the Euler-Maruyama Approximation

Let be a complete probability space with filtration , which satisfies the usual conditions, that is, the filtration is continuous on the right and contains all -null sets. Let denote the family of functions from to that are right-continuous and have limits on the left; is equipped with the norm , where is the Euclidean norm in , that is, . If is a vector or matrix, its trace norm is denoted by , while its operator norm is denoted by . Let denote by the family of all measurable, -valued random variables such that . For simplicity, we also denote by that .

In this paper, we consider the -dimensional neutral stochastic delay differential equations with Poisson jumps

where is a positive fixed delay, and , is a -dimensional standard Wiener process which is -adapted, and is a scalar Poisson process with intensity . Assume that and are independent of . Moreover the functions , , , .

The stochastic integral is defined in the Itô sense, and the integral version of (1) is frequently expressed as

We can now define the Euler-Maruyama approximate solution for NSDDEwPJs.

Given a step-size which satisfies for an integer , let for . Compute the discrete approximation by setting and performing

if , we have . Moreover, the increments are independent Gaussian random variables with mean 0 and variance ; are independent Poisson distributed random variables with mean and variance .

Let , , , with initial values on . The continuous Euler-Maruyama approximate solution is to be interpreted as the stochastic integral

It is not difficult to see that ; that is, and coincide with the discrete solution at the grid points.

In this paper, the following hypotheses are imposed on (1).

(H1) The Local Lipschitz Condition. There is a positive constant such that, for all with ,

(H2) The Constractive Mapping. There is a positive constant such that, for all

(H3) The Linear Growth Condition. There is a such that for all

Remark 1. Condition (H3) can be replaced by condition (H1).

In fact, from (5) and inequality , it is easy to obtain

similarly,

where

From (6), we obtain

where .

Recently we have studied the existence and uniqueness of solutions to neutral stochastic functional differential equations with Poisson jumps [21]. In an analogous way, we may establish the following existence and uniqueness conclusion that under assumptions (H1)-(H2), (1) has a unique solution on .

3. Lemmas and Corollaries

In this section, we first establish a few lemmas under local Lipschitz condition. For each , define the stopping times

(As usual we set .)

Lemma 2. Under (H1)-(H2), for any , there exists a constant independent of , such that

Proof. First, we prove the -moment of the exact solution of (1) is finite. For any , we obtain

From (2) and inequality

we get

By condition (H2) and inequality , then for any , we have

where . Thus for any ,

Similarly

By the Hölder inequality and (8), then for any , we get

Now, we use to denote that a generic constant may change between occurrences. Using the Burkholder-Davis-Gundy inequality for the two martingale integral terms, we have

Similarly, for the jump integral term, we have

By the above inequalities, we can obtain

that is,

where

The Gronwall inequality shows that

Then we can prove in the same way that the Euler-Maruyama approximate solution to (1) has the property that

So, we can obtain

by letting .

Lemma 3. Under (H1)-(H2),

where is a positive constant independent of .

Proof. For any , we have

By (4) and inequality

and (11) then we have

Using the Cauchy-Schwarz inequality, Fubini’s Theorem, and (8),

Then by the Doob martingale inequality and (8), we have

For the jump integral, we can transform to the compensated Poisson process

which is a martingale, and use the isometry

to obtain

Inserting (33)–(37) into (32) gives

Substitute the above inequality into (30), the result is

The Gronwall inequality shows that

This is the desired result.

Corollary 4. Under (H1)-(H2),

where is a positive constant independent of .

Lemma 5. Under (H1)-(H2), for any , there exists a constant independent of , such that

Proof. For any , there exists an integer such that . By the definition of and , we have

It is clear that and , and thus

Then applying the inequality and (8), we have

Then by Corollary 4 and the Lyapunov inequality [22]

Let ; the lemma is therefore complete.

4. Main Result

Theorem 6. Under assumptions (H1)-(H2), the Euler-Maruyama approximate solution (4) converges to the exact solution of the NSDDEwPJs (1) in the sense that

Proof. Let ; it is easy to see that

where is the indicator function of set .

Recall the Young inequality, for ; we have

Thus for any , we have

By Lemma 2, then

Similarly, the result can be derived for as

so that

Using these bounds along with

in (50) gives

Now we bound the first term on the right-hand side of (48). By the definition of and , we have

For ease of exposition, we abbreviate

Thus, for any ,

By the process of Lemma 5 and (45), we can get

By condition (H2) and (59), we have

By Cauchy-Schwarz inequality and condition (H1) and Lemma 5, we get

Similarly, by Burkholder-Davis-Gundy inequality and (8), we have

Then by (37) and condition (H1) and Lemma 5, we get

Taking (60)–(63) into (58), we obtain

Therefore

That is, for

where

Taking (55) and (66) into (48), we have

Given any , we can choose sufficiently small for

and then choose sufficient large for

and finally choose so that

Thus, . The proof is completed.

5. Numerical Examples

In this section, we present some numerical examples in support of our previous theoretical results.

Example 1. First we consider the following linear NSDDEwPJs:

First, we illustrate the strong convergence of the Euler-Maruyama method for NSDDEs with Poisson jumps. We choose and . To the best of our knowledge, there are not any analytical solutions available for Example 1. Therefore, we use the Euler-Maruyama method to compute an “explicit solution” with step-size in our experiments. We draw the numerical solution obtained from the Euler-Maruyama method with step-size in Figure 1. The data used in the figure is obtained by the mean square of data by 1000 trajectories; that is ; ; denotes the mesh point. From the figure, we can see that the exact solutions and numerical solutions match well.

Figure 1: The exact solution and the numerical method approximations for Example 1.

To show the strong convergence order of the Euler-Maruyama method, we apply the Euler-Maruyama method to Example 1. Then simulating the numerical solutions with 5 different step-sizes for , . The mean-square errors all measured at time are estimated by trajectory averaging. We plot our approximation to against on a log-log scale in Figure 2. For reference a dashed line of slope one-half is added. It clearly shows that the Euler-Maruyama method for Example 1 is convergent with order 1/2.

Figure 2: The convergence rate of the Euler-Maruyama method for Example 1.

Example 2. Consider the following nonlinear NSDDEwPJs:

We fix and for Example 2. The same as Figure 1, the data used in Figure 3 is obtained by the mean square of data by 1000 trajectories. In Figure 3, we show the numerical simulation of Example 2 by Euler-Maruyama method at step-sizes (upper), (middle), (lower).

Figure 3: The Euler-Maruyama method approximations for Example 2 with step-sizes (upper), (middle), and (lower).

Figure 4 also illustrates that the Euler-Maruyama method for Example 2 is convergent with order 1/2.

Figure 4: The convergence rate of the Euler-Maruyama method for Example 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported with funds provided by the National Natural Science Foundation of China (nos. 11226321, 11272229, and 11102132). The authors thank the two anonymous reviewers for their very valuable comments and helpful suggestions which improve this paper significantly.

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