Stability Analysis Of θ-Heun Method For Several Classes Of Stochastic Delay Differential Equations

Stochastic delay differential equations play an extremely important roles in daily life and various scientific fields.Most of them are difficult to obtain analytical solutions,so it is particularly significant to construct their numerical solutions and study their convergence and stability.The Heun method is a kind of efficient numerical method,and the θ-Heun method can be obtained by improving its drift term.Given its higher stability,it is meaningful to study the θ-Heun method.In this paper,for several kinds of random delay differential equations,the stability of θ-Heun method is discussed and analyzed,the main contents of this thesis as follows:1、The θ-Heun method is presented for solving stochastic delay differential equations.For one-dimensional linear random delay differential test equations,when their coefficients satisfy the condition that their solution is stable in a large range of random variables,this paper selects wiener increment in the random variable approximation metho d that obeys the two-point distribution,studies the T-stability of the θ-Heun method,and obtains the corresponding t-stability condition.Finally,the numerical test verifies the correctness of the conclusion.2、The paper gives various stability concepts of general nonlinear stochastic delay differential equation numerical methods.The mean square exponential stability,GMSstability and MS-stability of the θ-Heun method for solving the initial value problem of nonlinear stochastic delay differential equation are studied and the corresponding stability conditions of the method are obtained when the equation satisfies certain conditions.Finally,the numerical test verifies the correctness of the conclusion.3、The θ-Heun method is displayed for solving the stochastic delay differential equation with Poisson jump.According to the properties of random variable sequences,the almost sure exponential stability of θ-Heun method for solving Poisson jumps is studied,and then the almost sure exponential stability and MS-stability of θ-Heun method within a certain step size range is obtained when the equation satisfies certain conditions.Finally,the numerical test verifies the correctness of the conclusion.