Existence And Uniqueness Of The Solution And The Mean Square Stability Of Milstein-method For Several Stochastic Problems

In recent decades, as people in-depth study of economics, ecology, chemistry,engineering and other fields, practical problem appears many stochastic questionsmodel with fractional derivative or appears some time-related issues model thatrelated to current time and the previous. For these problems model, using integer-order stochastic differential equation simulations are unable to obtain the true phe-nomenon. Thus, many authors consider to promote the integer-order stochasticdifferential equations to fractional differential equations, or to discuss StochasticFunctional Differential Equations(SFDEs). This paper discusses the existence anduniqueness of the solution of the two types of Fractional Stochastic Differential Equa-tions (FSDEs) and prove the mean square stability of Milstein method of NonlinearNeutral Stochastic Delay Differential Equations(NSDDEs).In chapter one, we introduce the history that we study stochastic differentialequations and fractional differential equations, and () or are the researchstatus of fractional stochastic differential equations and stochastic delay differentialequations. Also, we introduce the main contents of this paper.In chapter two, we present some basic knowledge of fractional differential equa-tions and stochastic differential equations to prepare for the follow-up study.In chapter three, we discuss the existence and uniqueness of the fractionalstochastic differential equations. When>1/2, If certain conditions are met, thesolution of the fractional stochastic differential equations is existent and unique.In chapter four, we study on the existence and uniqueness of stochastic differen-tial equations with Caputo fractional derivative. When0<1/2, if the fractionalderivative of stochastic differential equations with Caputo satisfy certain conditions,then the solution is existent and unique.In chapter5, we are to study the numerical stability of Milstein-Method. Firstly,we build a Milstein Method for NNSDDEs, Then they obtain the mean-square sta-bility (MS-stability) of this presented method based on some sufficient conditions.The theoretical result is also confirmed by a numerical experiment.