Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which both the drift and the diffusion coefficients do not satisfy the global Lipschitz and linear growth conditions, especially the diffusion coefficients are highly non-linear growing. It is proved that the split-step theta (SST) method with θ∈[12,1] is strongly convergent to SDEPCAs under the local Lipschitz, monotone and one-sided Lipschitz conditions. It is also obtained that the SST method with θ∈(12,1] preserves the exponential mean square stability of SDEPCAs under the monotone condition and some condition on the step-size. Without any restriction on the step-size, there exists θ∗∈(12,1] such that the SST 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 presented to illustrate the analytical theory.