Two Classes Of Improved Explicit Numerical Methods For Solving Stochastic Differential Equations

With the development of science and technology,stochastic differential equations have become a very important mathematical model.Because it can describe problems more accurately in the natural,it is widely used in finance,neural network,biology and other fields.Because the right-hand functions of stochastic differential equations are very complex and few explicit solutions are available,how to construct efficient numerical methods is one of the hot fileds in the study of stochastic differential equations.However,most of the numerical methods require that the drift coefficient function and diffusion coefficient function of stochastic differential equations satisfy the global Lipschitz condition and the linear growth condition,which makes the application of the methods very limited.Therefore,it is very important to construct numerical methods under weakened conditions.Based on this idea,this paper construts two classes of explicit stochastic Runge-Kutta methods with order one in the mean-square sense.Firstly,this paper reviews the background and current research status of stochastic differential equations,introduces the research process of numerical methods of stochastic differential equations.Furthermore,this paper mainly discusses the results of two parts:In the first part,for a class of stochastic differential equations satisfying global monotonic conditions and superlinear growth conditions,an explicit balanced stochastic Runge-Kutta method with order one in the mean-square sense is constructed by controlling the drift coefficient function and diffusion coefficient function based on the balanced Milstein method.In the second part,for a class of stochastic differential equations satisfying global monotonic conditions and superlinear growth conditions,the projection idea is applied to the explicit stochastic Runge-Kutta method,and a class of explicit projection stochastic Runge-Kutta method with order one in the mean-square sense is constructed.The growth of solutions is controlled by projecting the numerical solutions of each step,so that the true solution of the original equation can be expressed more accurately.For the two classes of explicit stochastic Runge-Kutta methods proposed in this paper,the convergence of two methods is proved,and the effectiveness is verified by numerical examples.In the numerical examples,it is clear that both methods are of order one in the mean-square sense by comparing with the straight line with slope of 1.