Convergence And Stability Of Numerical Solutions For Two Types Stochastic Differential Equations With Piecewise Continuous Arguments

Convergence and stability are two important problems in numerical analysis.The convergence conditions and computational accuracy of numerical methods are the most important problems in convergence analysis.In stability analysis,the Lyapunov function method is an important method.In the absence of an appropriate Lyapunov function,the stability of the numerical solution is used to predict the stability of the exact solution.In addition,when the exact solution is stable,whether the numerical solution can maintain the stability of the exact solution is also a very important issue.This thesis focuses on the stochastic differential equations with piecewise continuous arguments(SDEPCAs)and the neutral stochastic differential equations with piecewise continuous arguments(NSDEPCAs)to study the properties of exact solutions and the convergence and stability of numerical methods.For SDEPCAs and their corresponding stochastic differential equations(SDEs)whose coefficients satisfy the global Lipschitz condition and linear growth condition,we study the equivalence of pth moment exponentially stability for SDEPCAs,SDEs and their corresponding Euler-Maruyama numerical method.The key to this research is to prove the convergence of the Euler-Maruyama method,the pth moment boundedness for the exact solutions of SDEPCAs,SDEs and the corresponding Euler-Maruyama numerical solutions,and the error of any two solutions in the sense of the pth moment;finally,the equivalence of stability for SDEPCAs,SDEs and their corresponding Euler-Maruyama method is established.For SDEPCAs whose coefficients satisfy the local Lipschitz condition and Khasminskii-type condition,this thesis constructs a truncated Euler-Maruyama method and studies the strong convergence and mean square exponential stability of the method.First,the pth moment(p ≥ 2)boundedness of the method is given,and then the strong convergence in the sense of (?)th moment((?) < p)and the order of convergence are obtained.Next,the sufficient condition for the truncated Euler-Maruyama method to maintain the exponential stability of the exact solution is discussed.When the coefficients do not satisfy the Khasminskii-type condition,but satisfy the generalized one-sided Lipschitz condition,this thesis continuous to study the strong convergence of the truncated Euler-Maruyama method.Since [t] is piecewise continuous,the higher power terms in the generalized one-sided Lipschitz condition are not easy to handle.Therefore,a segmented approach is adopted.First,the pth moment(p ≥ 2)boundedness of the exact solution is given;secondly,using the special properties of the truncated functions,we prove the boundedness of the pth moment(p ≥ 2)of the truncated Euler-Maruyama numerical solution;finally,the strong convergence and convergence order of the method are given.The conclusion of strong convergence of the truncated Euler-Maruyama method for SDEPCAs is extended to NSDEPCAs.For the NSDEPCAs whose coefficients satisfy the local Lipschitz condition and Khasminskii-type condition,the corresponding truncated Euler-Maruyama method is constructed.Since the neutral term contains [t],this method is implicit at the integer time node.Therefore,firstly,the solvability of implicit equations at integer nodes is analyzed;secondly,the boundedness of the pth moment of the exact solution and the numerical solution is given;and then the convergence and convergence order are obtained at a time T and a time interval [0,T ].