Two Kinds Of Split-step THETA-methods For Stochastic Pantograph Differential Equations

There is randomness in every real application. Stochastic models are playing a more and more important role in scientific and engineering, and has been gradually applied to many fields, such as economics, physics, finance, biology, communication, etc. So the study of stochastic differential equations (SDEs) becomes much more important in scientific research. However, only a few of stochastic differential equations have closed solutions and most of exact solutions have no explicit analytical formulae. Hence it is nec-essary and important to discuss some properties of numerical solutions in both theoretical and applied fields.Real application models are always effected by a history information. The delay term can be added to depict such problems in stochastic equations.In this thesis, we give a brief introduction for the application backgrounds and some present results of a kind of stochastic delay differential equations with vanishing delay, i.e.,stochastic pantograph equations (SPEs). After that, we investigate numerical properties of split-step θ-methods and one-leg split-step θ-methods for nonlinear non-autonomous SPDEs, respectively.For split-step θ-methods, the fundamental numerical properties concerning the strong convergence and mean-square stability are investigated. It is proved that the strong order is1/2 under a global Lipschitz condition,a linear growth condition and a polynomial con-dition for SPEs. It is also presented that under certain conditions, split-step θ-methods are mean-square stable when /q 1is a positive integer. Moreover, the numerical solutions are mean-square stable for all uniform mesh when θ= 1. Numerical experiments are given to verify our main conclusions.For one-leg split-step θ-methods, the strong convergence and mean-square stability are discussed. The same strong convergence order is presented under the same conditions in the case of split-step θ-methods. While, the numerical mean-square stability for SPEs is improved to 1/2<θ<1 when the underlying solutions are mean-square stable and 1/q is a positive integer. Numerical experiments are given to verify the conclusions.