Strong Convergence Of The Split-step One-leg ? Methods For Two Kinds Of Stochastic Differential Equations

Compared with deterministic differential equations,stochastic differential equa-tions take into account the influence of many random factors and are widely used in production practice.Since stochastic differential equations rarely have explicit solu-tions,it is very important to develop the numerical solution of stochastic differential equation.This paper is organized as follows:In the first chapter,we mainly introduce the research progress of the convergence of split step numerical methods for stochastic differential equations and the research status of the split-step one-leg ? methods(SSOLTM).In the second chapter,a class of methods,called improved split-step one-leg theta methods(ISSOLTM),converge strongly to the exact solution with the order 1 for SDEs with one-sided Lipschitz continuous drift coefficient and global Lipschitz continuous diffusion coefficient if the method parameter satisfies 1/2 ? ? ? 1.For 0 ? ? ?1/2,under the additional linear growth condition for the drift coefficient,the methods are also strongly convergent with the the order 1.Finally,the obtained results are supported by numerical experiments.In the third chapter,we present the split-step one-leg theta methods(SSOLTM)for stochastic delay differential equations.When the diffusion coefficient g(x,y)obeys the global Lipschitz condition in both x and y,but the drift coefficient f(x,y)satisfies one-sided Lipschitz condition and local Lipschitz condition in x and global Lipschitz condition in y,SSOLTM are shown to be strongly convergent for such SDDEs if the method parameter satisfies 1/2 ???1.Furthermore,we prove the strong conver-gence order is one half order if the local Lipschitz condition is replaced by a stronger condition.Finally,the obtained results are supported by numerical experiments.