Abstract
Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.
Highlights
In this paper we study the following nonlinear SDDE: dX (t) = f (X (t), X (t − τ (t))) dt (1)+ g (X (t), X (t − τ (t))) dW (t), for every t ≥ 0
To guarantee the almost sure stability of the unique solution to (1), we need the following assumption for the time-varying delay τ(t)
In what follows we introduce the result of almost sure stability of SDDEs (1)
Summary
In this paper we study the following nonlinear SDDE: dX (t) = f (X (t) , X (t − τ (t))) dt (1). Stability theory for numerical methods applied to stochastic differential equation (SDE) typically deals with mean-square behavior [1]. The mean-square stability analysis of numerical methods for SDDE has received a great deal of attention (see, e.g., [2, 3] and the references therein). Rodkina et al [7] studied almost sure stability of a driftimplicit θ-method applied to an SDE with memory. Discussed almost sure exponential stability of the EulerMaruyama (EM) method for the SDE with a constant delay and stochastic functional differential equation. We note that the two above schemes are all single-stage method; this paper studies the almost sure stability of a two-stage scheme named split-step backward Euler (SSBE) method [12, 13] applied to the nonlinear SDDE (1) with time-varying delay. The almost sure convergence of SSBE method has been investigated by Guo and Tao [14]; the main aim of this paper is to study the almost sure stability of the SSBE method applied to (1)
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