Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

Qiyong Li,Siqing Gan

doi:10.1155/2012/831082

Abstract

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 Δt = τ/m when 1/2 ≤ θ ≤ 1, and they are exponentially mean‐square stable if the stepsize Δt ∈ (0, Δt0) when 0 ≤ θ < 1. Finally, some numerical experiments are given to illustrate the theoretical results.

Highlights

Models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering
For SDDEs with jumps, most of the existing work is concerned about convergence property of numerical methods, see, for example, 16–19
Based on the above result, we are going to study the stability of numerical methods for 2.1

Summary

Introduction

Models that incorporate jumps have become increasingly popular in finance and several areas of science and engineering. For SODEs with jumps, the strong convergence and mean-square stability of some. For SDDEs with jumps, most of the existing work is concerned about convergence property of numerical methods, see, for example, 16–19. In 20 , 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 θ methods for nonlinear SDDEs with jumps. It is shown that the compensated stochastic θ methods inherit mean-square stability of the exact solution. Some numerical experiments are reported to illustrate the theoretical results

Stability of the Analytical Solution

Compensated Stochastic θ Methods for Nonlinear SDDEs with Jumps

Stability Analysis of the Numerical Solutions

E Xj 2 τ α2 β2 2λγ2

Numerical Examples