The Investigation Of Stable Numerical Solution For Stochastic Functional Differential Equations

Stochastic functional differential equations(SFDEs) can be viewed as generalizations of both deterministic functional differential equations(FDEs) and stochastic ordinary differential equations(SODEs).Since the random factor,together with the delay factor, is considred,SFDEs can always simulated the problems in practical truthfully.They have been widely used to model the corresponding systems in many scientific and engineering fields such as physics,biology,mechanics,finance and control.It is because analytical solutions are rare for stochastic differential equations that there is an increasing demand for numerical methods.As the existence of random factors,SDEs are more difficult and complex in constructing schemes rather than ordinary differential equations(ODEs).Among the researches on SFDEs,stability analysis is an important issue.At first,this paper deals with stability analysis of SFDEs in chapter 2.The p-th moment uniform asymptotic stability of the solutions is investigated by using Lyapunov functional and Razumikhin technique.A very general theorem is derived,by which we can get many corollaries,one of them is just the Ramumikhin-Mao theorem.As we all know,SFDEs are defined in the space of continuous functions.So only the solutions of the special functions such as stochastic delay differential equations(SDDEs) can be simulated.The investigation on numerical treatments of SDDEs is a new area.up to now,only few results have been presented.The main work is adapting the Euler-Maruyama method to solve the SDDEs,and studying the corresponding linear mean-square stability.But the convergence order of the method is only 0.5 and the accuracy is low.In chapter 3,we try to adapt the Milstein method with strong order 1 to solve the general linear SDDEs.and get the normal computational scheme,then the mean-square stability of the method is investigated by studying the moment property of a sort of stochastic multiple integrals. It is proved that the numerical method is mean-square stable under suitable conditions. The obtained result shows that the method preserves the stability property of the solved system.Furthermore,In chapter 4,the Milstein method is applied to nonliear SDDEs. When the analytical solution satisfies the conditions of mean-square stability,and if the drift term and diffusion term satisfy some restrictions,then the Milstein method is mean-square stable.At present,the research of numerical treatments fbr SDDEs is very difficult,one of the reasons is that the work in the area of SDEs is far less advanced than for deterministic differential equations.In the aspect of numurical schemes,the most common stochastic Taylor methods with derivatives of coefficient functions have been provided. to avoid finding the necessary derivatives,many derivative-free schemes such as Runge-Kutta(RK) methods are proposed.On the basis of the work by Burrage,R(o|¨)ssler,etc. In chapter 5,the paper constructs RK methods fbr strong solutions of the SDEs of It(?) type.We obtain the order conditions of the method by applying the colored rooted trees theory.Furthermore,we derive concrete schemes of two or three stages with order 1 by choosing new random variables.Furthermore,the mean-square stability of above methods is investigated.Finally,the accuracy of these methods is verified in numerical tests.When the diffusion term is equal to zero,SDDEs are degenerated to deterministic equations.In chapter 6,this paper deals with the nonlinear stability of extended general linear(GL) methods for solving multidelay integro- differential equations.The methods are composed of GL methods and compound quadrature rules.It is proved that the extended GL methods are asymptotically and globally stable under suitable conditions.