Research On Numerical Algorithms For Several Kinds Of Stochastic Differential Equations With Variable Delay

Stochastic variable delay differential equations can reasonably describe practical problems,so they are widely used in control science,biology,economics,population dynamics and other related fields.At present,there are few stochastic differential equations that can get analytical solution,so it is more important to solve the equations by numerical method.In practice,we study some characteristics of the algorithm to judge whether the numerical method is effective and reasonable.Due to some substantial difficulties,there are few research results on numerical methods for solving stochastic variable delay differential equations in existing literature.At present,only a few literatures have discussed the equation.On the one hand,the research on numerical methods of stochastic variable delay differential equations is limited to explicit or semi-implicit numerical methods;Therefore,this paper applies the fully implicit numerical method to stochastic variable delay differential equations and stochastic variable delay differential equations with Poisson jump,which not only extends the application range of the fully implicit numerical method,but also finds that the fully implicit method can be stable in a relatively large step size,and can improve the application in practical problems.On the other hand,another kind of stochastic variable delay differential equation is considered by using exponential Euler method,and it is concluded that this method can keep the analytical solution stable for(?)h>0.In this paper,we mainly solve several kinds of stochastic variable delay differential equations by numerical method.The full text is composed of the following four chapters.In chapter 1,the present situation of numerical analysis of stochastic delay differential equations is briefly described.In chapter 2,we study the convergence and stability of a class of stochastic variable delay differential equations by using the balanced methods.The results show that the balanced methods converges to the analytical solution of order 1/2?(??(0,1]).In addition,when the analytical solution is stable,both the strong balanced method and the weak balanced method can maintain their stability.Furthermore,numerical examples show that the theoretical analysis is reasonable and that the fully implicit balanced method is more stable than the explicit—Euler methods.In chapter 3,we study the convergence of stochastic variable delay differential equations with Poisson jumps in the mean-square sense by using the balanced methods.It is proved that the balanced method is convergence with strong order 1/2?(??(0,1]).Finally,the results of mean square convergence are verified by numerical simulation.In chapter 4,we investigate the convergence and stability of the exponential Euler method for semi-linear stochastic variable delay differential equations,and proved that the convergence order of is 1/2.The conditions for the analytical solution to be mean square stable are given,and it is proved that the exponential Euler method is mean-square stable for(?)h>0.Finally,a numerical example is given to verify that the theoretical results are reasonable.