Convergence And Stability For Fully Implicit Milstein Method Of Nonlinear Stochastic Delay Differential Equations

Models of stochastic delay diferential equation(SDDEs) often appear inmany felds of science such as fnance, biology, physics, chemistry, neural net-works, mechanical, environmental, etc.In recent decades, despite the numericalsolution of stochastic delay diferential equations study a lot, but mainly confnedto explicit and semi-implicit methods, so to study the fully implicit numericalmethods of stochastic delay diferential equations is necessary.This paper primar-ily construct a class of fully implicit Milstein method for nonlinear stochasticdelay diferential equations and analyze its convergence. Details are as follows:The frst chapter introduces the study history of explicit numerical methods,semi-implicit numerical methods and fully implicit numerical methods for therandom diferential equations and random delay diferential equations.Firstly, the second chapter uses the explicit Milstein method to write non-linear stochastic delay diferential equations corresponding discrete format. Thenthrough the analysis of the truncation error which is already acquired draws theconclusion that the format does not converge. Finally it uses the variable sub-stitution skills to obtain a fully implicit Milstein methods, and analysis of itsconvergence proved that the method is one-order convergent.The third chapter researchs the mean square stability (MS-stability) of thefully implicit Milstein methods, obtaining a sufcient condition for MS-stabilityof the method.The fourth chapter verifes the correctness of the theoretical results obtainedby numerical experiments.