Asymptotical mean-square stability of split-step theta methods for stochastic pantograph differential equations under fully-geometric mesh

Abstract

The paper deals with the numerical asymptotical mean-square stability of split-step theta methods for stochastic pantograph differential equations, which is the generalization of deterministic pantograph equations.  Instead of the quasi-geometric mesh, a fully-geometric mesh, widely used for deterministic problems, is employed. A useful technique, the limiting equation, for deterministic problems is also extended to stochastic problems based on Kronecker product. Under the exact stability condition, the  stability region of the split-step theta methods is discussed, which is an improvement of some existing results. Moreover, such technique is also available to stochastic pantograph differential equations with Poisson jumps. Meanwhile, compared with the destabilization of Wiener process, the stabilization of Poisson jumps is replicated by numerical processes. Finally, numerical examples are given to illustrate that our numerical stability condition is nearly necessary for stochastic problems.