Convergence And Stability Of Numerical Solutions For Two Classes Of Stochastic Differential Equations

Stochastic differential equations with piecewise continuous arguments(SDEPCAs)and stochastic differential equations with jumps have been applied into many fields.Many scholars have studied the properties of the numerical methods of SDEPCAs and ISDEs,there were also lots of results under the global Lipschitz condition.Since most of mathematical models do not satisfy the global Lipschitz condition,this thesis discusses the convergence and stability of the numerical methods for SDEPCAs and impulsive stochastic differential equations(ISDEs)under the non-global Lipschitz condition.This thesis reviews the background and history of SDEPCAs and ISDEs under the global Lipschitz condition,and summaries the current situations of the exact solutions and numerical solutions for SDEPCAs and ISDEs.The Monte Carlo Euler method of SDEPCAs is introduced.Under the super-linear growth condition,the pth moment boundedness of the exact and numerical solutions is studied.In addition,the convergence and the convergence rate of the Monte Carlo Euler method are also considered under the same condition.The implicit split-step θ methods are introduced in the third section.The pth moment boundedness of the split-step θ methods for SDEPCAs with polynomially growing coefficients is studied.The mean-square convergence of split-step θ methods and the convergence rate are analyzed.Furthermore,the sufficient conditions of the exponential stability in pth moment for the equations are given and the stability of the split-step θmethods is discussed.The pth moment convergence of numerical solutions for SDEPCAs with Poisson jumps is studied in the fourth section.The Tamed Euler method of the equations is given under the non-global Lipschitz condition.The pth moment boundedness of the Tamed Euler method is studied.Moreover,the convergence and the convergence rate of the Tamed Euler method are analyzed.Impulsive stochastic differential equations are considered in the fifth section.According to a special function,the impulsive stochastic differential equations are transformed to the impulsive stochastic differential equations without impulses and the meansquare stability of exact solutions of impulsive stochastic differential equations is studied.A modified Euler method is constructed and the strong convergence,and the exponential stability of the modified Euler method are analyzed.