Study On Numerical Methods For Solving Stochastic Differential Equations Based On Tree Theory

Stochastic differential equations (SDES) have been widely used in many fields, suchas economies, control science, ecology, etc, but it is very difficulty to find the analyticalsolution of a stochastic differential equation, so it becomes important to constructnumerical methods for solving stochastic differential equations.In Chapter1and2of this paper, we briefly introduce the background knowledge,basic concepts and theory about SDEs, colored rooted tree and stochastic Runge-Kuttamethods.In chapter3, for stochastic differential equations, according to colored rooted treetheory, this paper presents two classes of three-stage semi-implicit stochastic Runge-Kuttamethods for solving Stratonovich type stochastic differential equations, and analyzes theirmean square stability and mean square stability function. The stable regions of the abovemethods are also given, which are larger than that of the extant second order, third orderstochastic Runge-Kutta methods. The numerical simulation results show that the methodsabove have high accuracy.In chapter4, for stochastic delay differential equations, this paper studies theconvergence of the Heun method for solving the Itó form scalar stochastic delay differentialequations, and proves that the orders of convergence in the mean, in the mean-square and inthe mean square of the Heun scheme are2,1,1, respectively, if both the drift coefficientand diffusion coefficient satisfy the linear growth condition and global Lipschitz condition.