Abstract
This paper deals with the strong convergence and exponential stability of the stochastic theta (ST) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs) with non-Lipschitzian and non-linear coefficients and mainly includes the following three results: (i) under the local Lipschitz and the monotone conditions, the ST method with theta is an element of [1/2, 1] is strongly convergent to SDEPCAs; (ii) the ST method with theta is an element of (1/2, 1] preserves the exponential mean square stability of SDEPCAs under the monotone condition and some conditions on the step-size; (iii) without any restriction on the step-size, there exists theta* is an element of (1/2, 1] such that the ST method with theta is an element of (theta*, 1] is exponentially stable in mean square. Moreover, for sufficiently small step-size, the rate constant can be reproduced. Some numerical simulations are provided to illustrate the theoretical results. (C) 2021 Elsevier B.V. All rights reserved.
NSTL
Elsevier