Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations

Ling Zhang

doi:10.1186/s13660-017-1518-5

Abstract

The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order frac{1}{2} to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Highlights

Stochastic modeling has come to play an important role in many branches of science and industry
In this article we propose an explicit method to show that the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs) is proved to share the same stability for any step size by the property of logarithmic norm
4 Exponential stability in mean square we give the exponential stability in mean square of the exact solution and the exponential Euler method to semi-linear stochastic delay differential equations ( . )

Summary

Introduction

Stochastic modeling has come to play an important role in many branches of science and industry. Numerical solutions to SDEs have been discussed under the Lipschitz condition and the linear growth condition by many authors (see [ – ]). Higham et al [ ] gave almost sure and moment exponential stability in the numerical simulation of SDEs. Many authors have discussed numerical solutions to stochastic delay differential equations (SDDES) (see [ – ]). Zhang Journal of Inequalities and Applications (2017) 2017:249 we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the stability for any step size by the property of logarithmic norm. In Section , we obtain the convergence of the exponential Euler method to SLSDDEs under Lipschitz condition and the linear growth condition.

Preliminary notation and the exponential Euler method

Conclusions