Analytical And Numerical Study Of Some Classes Of Stochastic Functional Differential Equations

Stochastic functional differential equations (SFDEs) can be viewed as generalizationsof both deterministic functional differential equations (FDEs) and stochastic differentialequations (SDEs). Since the environmental noises and the retarded factor are considered, SFDEs can always simulated the problems in practical truthfully. They have been widely applied to model the corresponding systems in many fields such as chemistry, physics, ecology, medicine, finance, neural networks and control science. It is difficult to obtain the explicit solutions of general nonlinear SFDEs. Therefore, investigating appropriate numerical methods for the simulation to the solutions is very important in theory and in application. This paper aims at the analytical and numerical study of some classes of SFDEs.This paper is composed of seven parts.In chapter 1, a comprehensive survey of modern developments of analytical propertiesof SFDEs and their numerical methods is given. Furthermore, the main work of this paper is presented.In chapter 2, some elementary concepts of probability theory, stochastic processes and stochastic differential equation are introduced and some primary results which will be used in this paper are listed.In Chapter 3, a class of drift-implicit one-step schemes for numerically solving nonlinear neutral stochastic delay differential equations are established. The mean square convergence of the drift-implicit one-step schemes is studied. The relationship between consistence and convergence is obtained.In chapter 4, the split-step backward Euler (SSBE) method for numerically solving linear stochastic differential delay equations is constructed. The mean-square convergenceand MS-stability and GMS-stability of the SSBE method are investigated. It is shown that the SSBE method is convergent with order (?) in the mean square sense under appropriate conditions. Furthermore, some sufficient conditions to ensure that the SSBE method is MS-stable or GMS-stable are given. Finally, several numerical experiments are presented to illustrate the theoretical results in this chapter.In chapter 5. we consider nonlinear neutral stochastic pantograph equations given as follows:Firstly, it is shown that the Lipschitz condition and the linear growth condition guarantee the existence and uniqueness of the solution. Secondly, the semi-implicit Euler methods are convergent with order (?) in the mean square sense is proved. Finally, a numerical example is given to illustrate the theoretical order of convergence.In chapter 6, the analytical properties of nonlinear neutral stochastic pantograph equations on the infinite interval [0,∞) are studied. Estimates for the up bound of the p-th moment Lyapunov exponent and the sample Lyapunov exponent of the analytical solutions are obtained.In chapter 7. the convergence of semi-implicit Euler methods with linear interpolationfor nonlinear stochastic pantograph equations is investigated. it is proved that these methods are convergent with order (?) in the mean square sense. At last, a numericalexample to illustrate this analytical result is presented.