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

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