The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments

Abstract

In this paper we consider stochastic differential equations with piecewise constant arguments. These equations describe hybrid dynamical systems, that is, combinations of continuous and discrete systems. Our aim is to establish the mean square convergence of the Euler–Maruyama approximate solution under quite general conditions, that is, the global Lipschitz condition and the linear growth condition, which guarantee the existence and uniqueness of the true solution. Then we show that the initial equation is exponentially stable in mean square if and only if, for some sufficiently small step-size Δ, the Euler–Maruyama method is exponentially stable in mean square. The stability study does not involve the Lyapunov functions nor functionals. It should be pointed out that stochastic differential equations with piecewise constant arguments can be regarded as a class of stochastic differential equations with multiple time-dependent delays which can be found in the literature. However, in the convergence analysis of the Euler–Maruyama method of these equations, it is required that the delay functions satisfy Lipschitz continuity condition. In the present paper, the delay functions do not satisfy that condition, so it represents an extension of results from the papers Mao (2003, 2007), using the technique similar to that from the first paper.