Convergence And Stability Of ?-Heun Methods For Stochastic Delay Differential Equations

The stochastic delay differential equation is a kind of model developed on deterministic delay differential equations and stochastic ordinary differential equations,which is similar to the actual problem.This kind of equations is widely used in many fields of science,such as population problem,memory materials or memory systems.In recent years,there has been increasing interest in the study of functional differential equations containing memory or after-effect.On the one hand,the stochastic delay model can show delay or after-effect,while the deterministic model requires that the parameters are known.On the other hand,in reality,there are often noise or interference terms,and the stochastic delay model provides a model that is similar to reality than the deterministic model.Therefore,the stochastic delay model has attracted more and more attention,and stochastic delay differential equation has become an important mathematical model.Based on the above reasons,in this dissertation,the related problems of stochastic delay differential equations are studied.The stochastic delay differential equation mainly includes the study of linear stochastic delay differential equations and nonlinear stochastic delay differential equations.In most cases,there is no theoretical solution to stochastic delay differential equations.Even if a small part can give a theoretical solution,it is not suitable for calculation in practice.Therefore,it is necessary to study stochastic delay differential equations by numerical method.The analysis of the numerical method of stochastic delay differential equations is based on the numerical analysis of deterministic delay differential equations and the numerical analysis of stochastic ordinary differential equations.Most of the numerical methods of stochastic delay differential equations are based on Euler methods and ?-methods.Based on the existing results of stochastic delay differential equations,this dissertation gives the research of ?-Heun method.At present,as far as we know,most of the numerical methods of stochastic delay differential equations are studied under the condition of global Lipschitz and linear growth.However,because most of the equations do not satisfy the global Lipschitz condition well,it is necessary to discuss the ?-Heun method of stochastic delay differential equations under non-global Lipschitz conditions.On this basis,the convergence and stability of ?-Heun method for this kind of stochastic delay differential equations are analyzed.