Numerical Methods And Their Theory For Several Classes Of Stochastic Differential Equations With Time-variable Delays

Stochastic modelling has come to play an important role in many branches of science and engineering.More and more scholars begin to study the stochastic differential equation.With the deepening of the research in recent years,various classes of stochastic differential equations began to get the attention of scholars,for example,stochastic differential equations with constant delay,stochastic differential equations with time-variable delay,poisson jump-diffusion stochastic differential equations,stochastic differential equations with Maikovian switching.However,the exact solutions of most stochastic differential equations with delays and other types of random process cannot be given explicitly.Thus,it’s becoming increas-ingly important to research the numerical solution of stochastic differential equation.In this work,we consider the analytic solutions and the numerical algorithms of several classes of Ito stochastic differential equations with time-variable delay and focus on studying the convergence and stability.In Chapter 1,in view of the stochastic differential equation widely used in various fields,we briefly cite several stochastic models.Morevoer,we review the research develop-ment of stochastic differential equations and its numerical algorithm.Also,we introduce the main content and the research significance of this work,and introduce some commonly used notations,definitions and basic theory.In Chapter 2,we consider nonlinear stochastic functional differential equations with piecewise continuous arguments(SFDEPCAs).The underlying one-leg θ-methods are adapted to solve SFDEPCAs.Under the global Lipschitz condition and the strongly linear growth condition,it is proved that the adapted stochastic one-leg θ-methods are convergent with strong order 1/2.Moreover,we give a mean-square exponential stability criterion of the adapted one-leg θ-methods.At last,some numerical experiments are given to illustrate the computational effectiveness and the theoretical results of the induced methods.Chapter 3 deals with numerical solutions for nonlinear stochastic differential equa-tions with jump-diffusion and piecewise continuous arguments under the one-sided Lips-chitz condition.The compensated split-step balanced methods(CSSBB)for the equations are suggested.It is proved that the solutions of CSSBB methods are convergent using the continuous-time approximation rather then discrete approximation which is usually ana-lyzed in a class of balanced methods.Also,we give the criterion that when the CSSBB methods satisfy the corresponding conditions,the mean-square exponentially stability of the numerical solutions for CSSBB methods preserves the mean-square exponentially sta-bility of the analytical solutions.At last,two numerical examples show that CSSBB methods are more effective than those methods which are without balanced term.In Chapter 4,we focus on the backward Euler-Maruyama method for nonlinear hybrid SDEs with time-variable delay.Under the local Lipschitz condition and polynomial growth condition,the equation has a highly nonlinear problem and the effect of time-varying delay.Therefore,it’s always difficult to get the strong convergence properties of the numerical solution.By citing several lemmas and the technique for dealing with time-varying delay,it is proved that the backward Euler-Maruyama method is strongly convergent.Moreover,applying the continuous and discrete semi-martingale convergence theorems proves that nu-merical methods preserve the almost surely exponential stability of the analytic solutions to highly nonlinear hybrid SDEs with time-variable delay under the appropriate conditions.A numerical experiment is given to illustrate the computational effectiveness and the theoreti-cal results of the method.Chapter 5 is concerned with asymptotical boundedness and moment exponential sta-bility for nonlinear stochastic neutral differential equations with time-variable delay and markovian switching.Two main theorems are given respectively.First,we give the analysis of the Lyapunov theory for nonlinear hybrid neutral stochastic differential delay equations.Based on the first criterion and the property of M-matrices,the second criterion shows the asymptotic boundedness and the moment exponential stability are tenable when we give the general conditions depended on coefficients of the nonlinear hybrid neutral stochastic dif-ferential delay equations.Under the analysis,we can find that even if there is a certain state not asymptotically bounded or exponentially stable,the systems still keep asymptotically bounded or exponentially stable.At last,an example is also given to illustrate the theoretical results.In the last chapter,this work is summarized,and we describe the prospects for the future.