Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

Abstract EN

A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes.

A general framework for mean-square convergence of the methods is provided.

It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error.

The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods.

It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2.

Numerical experiments illustrate the theoretical results.

It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.