Stability Of Analytic Solution And Convergence Of Numerical Solution For Stochastic Pantograph Differential Equations

Stochastic delay differential equations (SDDEs) play important roles in scienceand industry, and applied widely in many fields, such as finance, economics, biology,control science. Since the explit solutions of SDDEs can hardly be obtained,investigating appropriate numerical methods and researching the convergence ofnumerical method is very important. Stability is another important property ofsolution, it reflects the influence of the initial value, coefficient for the solution ofequations. Therefore, stability of solution also has very important theoreticalsignificance and application value.Stochastic pantograph differential equations (SPDEs) are special SDDEs. Thispaper main discusses the convergence of the numerical solution for nonlinearSPDEs and stability of analytical solution for the m-dimensional linear SPDEs.Firstly, the convergence of the numerical solution for nonlinear SPDEs isstudied. Apply the Milstein method to nonlinear SPDEs, we can obtain the numericformat of Milstein method, then we proved Milstein method is mean-squareconvergence under the conditions of linear growth and global-Lipschitz.Secondly, the stability of analytical solution of the m-dimensional linear SPDEsis studied. We discuss the mean-square polynomial stability and almost surelypolynomial stability respectively, give the sufficient conditions of mean-squarepolynomial stability and almost surely polynomial stability of the analytic solutionof m-dimensional linear SPDEs.