Numerical Analysis For Several Classes Of Stochastic Delay Differential Equations

Stochastic modeling has come to play an important role in many branches of sci-ence and industry. Such models have been used with great success in a variety of appli-cation areas, including biology, epidemiology, mechanics, economics and finance. Moststochastic differential equation are nonlinear and cannot be solved explicitly. Therefore, itis important to obtain some approximations by numerical method. In general, we presentthe property of exact solution by using that of approximations. The main purpose ofthis paper is to investigate the convergence and the exponentially stable in mean squareof the Euler method to stochastic delay differential equations and stochastic differentialequations with piecewise continuous arguments.In this thesis, we show the application background of stochastic delay differentialequations and stochastic differential equations with piecewise continuous arguments. Andwe give the history and development worth to study numerical solutions.For stochastic differential equations with piecewise continuous argu-ments(SEPCAs), this paper is to investigate the strong convergence of the Eulermethod to stochastic differential equations with piecewise continuous arguments. Firstly,it is proved that the Euler approximation solution converge to the analytic solutionunder local Lipschitz condition and the bounded pth moment condition. Secondly, theEuler approximation solution converge to the analytic solution is given under localLipschitz condition and the linear growth condition. Then we consider the same resultwithout the linear growth condition. That is, the convergence of numerical solutionsto stochastic differential equations with piecewise continuous arguments under localLipschitz condition and the monotone condition is established. Finally, an exampleis provided to show which is satisfied with the monotone condition without the lineargrowth condition. And some numerical experiments are given.The convergence in probability of the Euler method to stochastic differential e-quations with piecewise continuous arguments (SEPCAs) is discussed. The classicalKhasminskii-type theorem gives a powerful tool to examine the global existence of solu-tions for stochastic differential equations without the linear growth condition by the useof the Lyapunov functions. However, there is no such result for stochastic differential equations with piecewise continuous arguments. Firstly, this paper shows stochastic dif-ferential equations with piecewise continuous arguments which have nonexplosion globalsolutions under local Lipschitz condition without the linear growth condition. Then theconvergence in probability of numerical solutions to stochastic differential equations withpiecewise continuous arguments under the same conditions is established. Finally, anexample is provided to illustrate our theory.The strong convergence and the exponentially stable in mean square of the exponen-tial Euler method to semi-linear stochastic differential equation with piecewise continuousarguments (SLSEPCAs) is showed. Firstly, we show that the exponential Euler approx-imate solution converge to the analytic solution with the strong order12to semi-linearstochastic differential equation with piecewise continuous arguments under Lipschitz con-dition and linear growth condition. Then we give the exponentially stable in mean squareof the exact solution to semi-linear stochastic differential equation with piecewise con-tinuous arguments by using interval with integral end-points method and definition oflogarithmic norm. the exponential Euler method to semi-linear stochastic differential e-quation with piecewise continuous arguments is proved to share the same stability for anystep size by property of logarithmic norm. Finally, two example are provided to illustrateour theory.For semi-linear stochastic delay differential equations (SLSDDEs), the strong con-vergence and the exponentially stable in mean square of the exponential Euler methodis considered. we show that the exponential Euler approximate solution converge to theanalytic solution to semi-linear stochastic delay differential equations under Lipschitzcondition and linear growth condition. Then we give the exponentially stable in meansquare of the exact solution to semi-linear stochastic delay differential equations by usingdefinition of logarithmic norm and direct prove method. the exponential Euler methodto semi-linear stochastic delay differential equations is proved to share the same stabilityfor any step size by property of logarithmic norm. Finally, two example are provided toillustrate our theory.