Convergence and stability of Euler method for impulsive stochastic delay differential equations

Abstract

This paper deals with the mean square convergence and mean square exponential stability of an Euler scheme for a linear impulsive stochastic delay differential equation (ISDDE). First, a method is presented to take the grid points of the numerical scheme. Based on this method, a fixed stepsize numerical scheme is provided. Based on the method of fixed stepsize grid points, an Euler method is given. The convergence of the Euler method is considered and it is shown the Euler scheme is of mean square convergence with order 1/2. Then the mean square exponential stability is studied. Using Lyapunov-like techniques, the sufficient conditions to guarantee the mean square exponential stability are obtained. The result shows that the mean square exponential stability may be reproduced by the Euler scheme for linear ISDDEs, under the restriction on the stepsize. At last, examples are given to illustrate our results.