Semi-analytical Split Milstein Method For Solving Stochastic Differential Equations

Most stochastic differential equations are difficult to find explicit expression of the solution,so numerical calculation has become an important research method.However,for stochastic differential equations with super-linear growing coefficients,the classical explicit scheme does not have convergence.The improved explicit methods,including tamed method,truncated method,projected method,adaptive time-stepping method,etc.,are applied to solve superlinear stochastic differential equations to obtain convergence in L~psense.Different from these improved explicit schemes,this thesis constructs a semi-analytical splitting Milstein method to solve the stochastic differential equations with super-linear growing coefficients.First,the superlinear stochastic differential equation is rewritten as a system of per-turbed equations,which consists of an ordinary differential equation and a new(with linearly increasing coefficient)stochastic differential equation.Here the explicit expres-sion of the ordinary differential equation,whose right-hand term is the superlinear part of the drift coefficient of the original stochastic differential equation,can be obtained an-alytically.Next,on each discrete interval,we find the analytical solution of the ordinary differential equation at the right endpoint of the interval first.Then taking this as the initial value,we use the explicit Milstein scheme to find the value of the new stochastic differential equation at the right endpoint of the interval and complete a one-step update here.Finally,we can find approximate values at different moments through multi-step it-erative updates.This thesis proves that the mean-square convergence order of the method is 1.This thesis also discusses some long-time properties of the method,including the mean-square boundedness of the numerical solutions,the exponential stability,and the convergence of the numerical stationary distribution.The correctness and validity of the theoretical results have been verified by several numerical experiments.