Convergence And Stability Of Numerical Solution Of Neutral Stochastic Differerntial Equations With Delay

Neutral delay stochastic differential equation(NSDDEs)can depict dependent on both the past and the present status and dependent on the state of the rate of change in the past a period time of the stochastic process,widely used in the economic,financial,and radio communication and other scientific research and engineering fields.However,since the analytical solution of this type of equation is troublesome to acquire,the study of the numerical solution is increasingly stressed on focus and the study of the convergence and stability of the numerical solution is an essential content of this work.This paper mainly considers the convergence and stability of NSDDE numerical solution whose delay term is an unbounded variable(hereinafter referred to as NSDDE with unbounded time-dependent delay).Chapter 1,namely the introduction,mainly introduces the research background of NSDDE with unbounded time-dependent delay,the previous research results of the convergence and stability of the numerical solution of the equations,the main research content and prospects.Chapter 2 summarizes the main basic definitions and inequalities used.Chapter 3,under the local Lipschitz continuous condition and the Khasminskii type condition the global existence and uniqueness of the analytical solution of NSDDE with unbounded time-dependent delay is verified and the modified truncated EM method is strongly convergent.Then,presented a numerical example.Chapter 4,under the local Lipschitz continuous condition and the improved Khasminskii type condition the asymptotic exponential stability of the NSDDE analytical solution and the modified truncated EM numerical scheme with unbounded time-dependent delay is proved.Then I present a numerical example to verify the asymptotic exponential stability of the numerical scheme.Chapter 5,the modified truncated EM method is improved.Under local Lipschitz continuous condition and Khasminskii type condition the improved modified truncated EM method is strongly convergent.Finally,the advantages of the method are summarized by comparing with the modified truncated EM numerical scheme.