Efficient Algorithm Design And Stability Analysis For Stochastic Delay Differential Equations

Compared with the deterministic model,the stochastic model can describe the problems in real life more realistically and reliably.Therefore,the stochastic model has a wide application in scientific theory and production practice,for example: economics,biology,medicine and finance and so on.Unfortunately,in most cases,it is impossible to find the analytic solution directly for stochastic models,so it is important to study the numerical solutions of SDDs and SDDEs,and it is essential to construct the numerical method with high precision and simple format.Firstly,this thesis introduces the background knowledge and research significance related to SDEs and SDDEs,and lists some inequalities and Lemmas that need to be used in the future.Then for the numerical solution of SDEs and SDDEs,in order to avoid the problem of complicated derivative terms and large computation in traditional difference methods,this thesis introduces a new variable based on the general stochastic RK method,and constructs an explicit stochastic RK methods with strong order-1.Furthermore,this thesis not only uses the constructed explicit strong order-1 stochastic RK methods to write the discrete schemes corresponding to nonlinear SDEs and nonlinear SDDEs,but also solves the numerical solutions by MATLAB.Then,this paper focuse on the mean square stability and almost sure exponential stability of the numerical solution.And the paper obtain that for general nonlinear SDEs and SDDEs,when the drift term and diffusion term of the equation satisfy certain conditions,if the theoretical solution of the system is mean-square stable or almost necessarily exponential stable,the numerical solution obtained by this method can maintain the two stability.Finally,this thesis gives three specific equations,namely a nonlinear SDEs,a linear SDDEs and a nonlinear SDDEs.They are solved by using the explicit strong order-1 stochastic RK methods.Numerical experiments verify the correctness of the theoretical results obtained.