The θ-method Study Of Several Classes Of Stochastic Functional Differential Equations

Stochastic delay differential equations have been widely used in science and engineering fields.But it is difficult to obtain the analytic solutions of equations.Thus the development of numerical methods has become an important content.Among them,the study of mean square stability is very important in the numerical algorithm.Therefore,in this paper,the mean square convergence and stability of numerical methods for three classes of stochastic functional differential equations are studied:In chpter 1,we briefly introduce the research background,significance and development status of stochastic delay differential equations.At the same time,it introduces the structure of the thesis.In chpter 2,the mean square stability of the split-step Milstein schemes for stochastic differential equations has been proved,and a numerical example has been given to support the theoretical results.In chpter 3,we concentrate on the numerical stability of linear stochastic delay integro-differential equations(SDIDEs).We establish the split-step θ-scheme and the stochastic linear theta method to solve linear SDIDEs and to discuss its stability.It is proved that the two classes of the theta methods with θ∈[0,1/2]can recover the exponential mean square stability when some restrictive conditions on stepsize and the drift coefficient are imposed,but for θ∈(1/2,1],these two classes of numerical schemes can reproduce the exponential mean square stability unconditionally.In chpter 4,we give a series of lemmas for the nonlinear neutral stochastic delay differential equations,and prove the stability in mean-square sense of the split-step θ-method.Finally,linear and nonlinear equations are used to illustrate the mean square stability of the split-step θ-method.