Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 810160, 11 pages
http://dx.doi.org/10.1155/2015/810160

Research Article

A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes

Ying Du and Changlin Mei

Department of Statistics, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received 29 September 2014; Revised 13 January 2015; Accepted 14 January 2015

Academic Editor: Chin-Chia Wu

Copyright © 2015 Ying Du and Changlin Mei. 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.

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-step method is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.

1. Introduction

Stochastic differential equations (SDEs) for jump-diffusions arise in a variety of practical areas and have successfully been used to describe unexpected and abrupt changes in the present structure (for an overview, see [1, 2]). Generally, SDEs for jump-diffusions cannot be solved explicitly. Therefore, constructing some forms of numerical solution and studying their properties have received a great deal of attention. When the jump magnitude is deterministic, Higham and Kloeden [3–5] studied the convergence and stability of the numerical solutions of the stochastic differential equations with jumps. Wang et al. [6] proved that the semi-implicit numerical solutions of stochastic delay differential equations with jumps are convergent to their corresponding analytical solutions. Jiang et al. [7] investigated the Taylor approximation for stochastic delay differential equation with jumps (SDDEJs). Bao et al. [8] obtained the convergence rate of the Euler-Maruyama method for SDDEJs under the local Lipschitz condition. Furthermore, some researchers [9–11] extended the constant delay in SDDEJs to variable delay .

Chalmers and Higham [12] extended the results in [4] to the case where the jump magnitudes are random. Stochastic differential equations with compound Poisson processes have been commonly used in mathematical finance and covering a wide range of finance models [13–16]. Considering the aftereffect of the past state, Jiang et al. [17] proposed a semi-implicit Euler numerical method for stochastic differential delay equations with Poisson driven jumps of random magnitudes. Furthermore, Mao [18] studied the stochastic differential equations with variable delays and random jump magnitudes (SDEVDRJMs) which are of the form

where is a variable delay, is an -dimensional standard Wiener process, is a scalar Poisson process with intensity , are independent and identically distributed random variables representing jump magnitudes, the drift coefficient and the jump coefficient are -valued, the diffusion coefficient is -valued for , and is starting delay condition function. Moreover, it is generally assumed that, for some , there exists a constant such that . Mao [18] proposed a semi-implicit Euler numerical method for SDEVDRJMs and proved that the numerical solutions converge to their analytical solution both in mean-square and in probability.

However, as pointed out by Chalmers and Higham [12], the semi-implicit Euler method for stochastic differential equations with random jump magnitudes is stable in mean-square but lose the A-stable result for constant jump magnitude. Here, A-stable property means that “problem stable method stable for all time step size .” Furthermore, it seems that there is little literature on the study of the stability of numerical methods for SDEVDRJMs.

Our work is motivated by [3], where compensated split-step backward Euler method is introduced and proved to have a satisfactory stability property. In this paper, a compensated split-step method is proposed to construct the numerical solutions of SDEVDRJMs, and the mean-square convergence of the numerical solutions is proved. Furthermore, the mean-square stability of the analytical solutions and the numerical solutions is investigated via a test equation; the results demonstrate that the proposed numerical method possesses not only the mean-square stability but also the A-stability.

2. Compensated Split-Step Numerical Solutions for SDEVDRJMs

Let be a complete probability space with the filtration satisfying the usual conditions that is right-continuous and contains all -null sets. Let be the Euclidean norm in and let be the scalar product of vectors . Denote with being a -dimensional real random variable.

We firstly introduce the following assumptions for the establishment of the convergence and stability of the proposed numerical solution of the SDEVDRJMs in (1).

(i) The time delay satisfies

For , there exists positive constant such that

(ii) Global Lipschitz condition: there exists a positive constant such that, for any , , (), letting ,

(iii) Quadratic growth condition: for any (), there exists a positive constant such that

(iv) For , , there exist positive constants and such that and .

Theorem 1. Under the assumptions (i) and (iii), there exist positive constants and , such that the solution of the SDEVDRJM in (1) satisfies

for any .

Proof. From (1) and the linear growth condition, it can be obtained that, for any ,

Let and . We have

Using Gronwall inequality, we obtain

Furthermore,

with .

Using the compensated Poisson process which is a martingale, we can rewrite

where

Obviously, still satisfies the global Lipschitz condition and the linear growth condition. However, this equation is complicated and its analytical solution is in general hard to be obtained. In what follows, we use the compensated split-step method to construct its numerical solution.

For a given constant time step size , let

Then the compensated split-step numerical solutions of the SDEVDRJM is of the form

where is the numerical approximation to , , and for with and being the integer part of .

3. Convergence of the Numerical Solutions

In this section, we will prove that the above numerical solutions converge in mean-square to the true solution of the SDEVDRJM with the approximate rate of .

Replacing the numerical approximations with the exact solution values on the right-hand side of (14a) and (14b), we obtain

Then, the local error of the compensated split-step method for the approximation of the solution of (1) is

and the global error is

Obviously, is -measurable because both and are -measurable.

Theorem 2. Under the assumptions (i)–(iv), there exists a constant such that, for any , the local error of the compensated split-step method satisfies

where the constants and are independent of .

Proof. From (14a), (14b), (15a), and (15b), we have

Thus

To estimate , let us consider the following five possible cases.
(1) If , then

From Theorem 1, it is obtained that

(2) If , then

Thus .
(3) If and , then

Therefore,

(4) If and , then

Thus

(5) If and , then

Thus

Combining the above five cases, it can be concluded that there exists a constant such that .
Similarly, it can be proved that there exists a constant such that . Therefore,

From (15a) and Theorem 1, we have, for any , that there exists a constant such that

Hence,

Moreover, it is known from Theorem 3.4 in [12] that

Thus,

Summing up the above conclusions, we can obtain that, for any , there exists a constant such that

Otherwise, for any , there exists a constant such that

Theorem 3. Under assumptions (i)–(iv), there exists a constant such that, for any , the global error of the compensated split-step method satisfies

where is independent of .

Proof. From the definitions of and , we have

where

From (14a), (15a), and assumption (ii) it can be obtained that, for any ,

and, for any ,

Since , , , and , we known from assumption (ii) that

where , . Hence

From Theorem 2, we have

Combining the above conclusions, we have

Additionally, when , . From (45), it can be obtained that the following items are true.
(1) If and , we have

(2) If and , we have

(3) If , we have

Combining the above three cases, it can be obtained that

where .

4. Stability of the Analytical and Numerical Solutions for the SDEVDRJM

In this section, we will discuss the stability of the analytical solutions of the SDEVDRJM and the numerical method introduced in Section 2.

Consider the following scalar test equation:

where and are real constants. Define , where and are the jump times.

4.1. Mean-Square Stability of the Analytical Solutions

In what follows, we give some sufficient conditions on the stability property of the analytical solutions of (50).

Theorem 4. Assume that the constants , , , , and and the random variable satify

Then solution of (50) is mean-square stable. That is,

Proof. For any and , it follows from Itô formula that

Taking the expectation yields

Let , , and . It is obtained that

From (51), we know . Furthermore, according to lemma 1.1 in [19], there exist and such that

Thus, when .

4.2. Mean-Square Stability of the Numerical Solutions

Applying the compensated split-step method to the test equation yields

Note that a numerical method is said to be mean-square stable (MS-stable) if there exists a constant such that for all and a numerical method is said to be general mean-square stable (GMS-stable) if holds for every time step size . For notational simplicity, let

Theorem 5. Assume that condition (51) holds. Then we have the following conclusions.
(i) If and or and , then the compensated split-step method for (50) is GMS-stable.
(ii) If and , then there exists a constant such that, for any , the compensated split-step method for (50) is MS-stable.
(iii) If and , then there exists a constant such that, for any , the compensated split-step method for (50) is MS-stable.
(iv) If , then there exists a constant such that, for any , the compensated split-step method for (50) is MS-stable.
Here,

and .

Proof. From (57a), if , we have

Then, it can be obtained that

From (57b), it is true that

Hence,

Therefore,

where

Obviously, . Then, by recursive calculation, we conclude that if and only if

that is,

Let

Then, we obtain the following conclusions.

(1)If , then (68) can be written as

 Furthermore,

(a)if , we have for all , which means that (66) is true for all ;(b)if , we have for any . Thus inequality (66) is hold for any

 which implies that (66) holds for such .(2)If , then we have the following:

(a)if , we have for all , which means that (66) is true for all ;(b)if and , we have for all , which means that (66) is true for all ;(c)if and , we have for any .

(3)If , we have

 Therefore, we have for any .

The above results show that the compensated split-step method is of the mean-square stability for the test equations. Furthermore, it is obtained that when the parameters , , , , and make the problem stable, there exists such that the method is stable for any time step size , which implies that the proposed numerical method is of the A-stable property.

5. Conclusions

In this paper, A compensated split-step method is formulated to numerically solve SDEVDRJMs. It is proved that the numerical solutions converge in mean-square to their analytical solutions with such a rate that is arbitrarily close to . Furthermore, the stability of the compensated split-step method is investigated via a test equation and the results show that the proposed numerical method is of not only mean-square stability but also the A-stable property at least for the test equation. The results demonstrate that the proposed compensated split-step method is of satisfactory performance for numerically solving SDEVDRJMs.

Conflict of Interests

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

Acknowledgments

The authors thank the anonymous referee and the editors for their careful reading of this paper and for their valuable comments. This research is funded by the National Science Foundation of China (no. 11201365).

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