Convergence and stability of stochastic theta method for nonlinear stochastic differential equations with piecewise continuous arguments

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 θ∈[1/2,1] is strongly convergent to SDEPCAs; (ii) the ST method with θ∈(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 θ∗∈(1/2,1] such that the ST method with θ∈(θ∗,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.