Convergence and stability of the compensated split-step theta method for stochastic differential equations with piecewise continuous arguments driven by Poisson random measure

Abstract

This paper deals with the numerical solutions of stochastic differential equations with piecewise continuous arguments (SDEPCAs) driven by Poisson random measure in which the coefficients are highly nonlinear. It is shown that the compensated split-step theta (CSST) method with θ ∈ [ 0 , 1 ] is strongly convergent in p th( p ≥ 2 ) moment under some polynomially Lipschitz continuous conditions. It is also obtained that the convergence order is close to 1 p . In terms of the stability, it is proved that the CSST method with θ ∈ ( 1 2 , 1 ] reproduces the exponential mean square stability of the underlying system under the monotone condition and some restrictions on the step-size. Without any restriction on the step-size, there exists θ ∗ ∈ ( 1 2 , 1 ] such that the CSST method with θ ∈ ( θ ∗ , 1 ] is exponentially stable in mean square. Moreover, if the drift and jump coefficients satisfy the linear growth condition, the CSST method with θ ∈ [ 0 , 1 2 ] also preserves the exponential mean square stability. Some numerical simulations are presented to verify the conclusions.