Two Classes Of Improved Explicit Stochastic Runge-Kutta Methods

The theory of stochastic differential equations(SDEs)is widely used in the fields of ecology,dynamics,risk management and so on.However,since the complexity of SDEs,most nonlinear SDEs cannot be solved explicitly,expect for some linear ones.Therefore,the numerical methods of solving SDEs become important in the application of SDEs,especially the construction of efficient numerical methods.In general,when studying the convergence of numerical methods for SDEs,the drift coefficient and diffusion coefficient of most SDEs are required to satisfy both the global Lipschitz condition and the linear growth condition.However,in the real world,many models characterized by SDEs do not satisfy these two conditions.Therefore,the construction of numerical methods to solve this type of SDEs has become a hot issue.Therefore,in this paper the explicit stochastic Runge-Kutta method is improved.Under the non-global Lipschitz condition and the superlinear growth condition,two types of convergent explicit stochastic Runge-Kutta methods are constructed.First,for SDEs whose drift coefficients and diffusion coefficients satisfy monotonic and superlinear growth conditions,we improved a type of 1-stage stochastic Runge-Kutta method and constructed explicit 1-stage stochastic Runge-Kutta method with varying coefficients.In the sense of p-moment,we proved this method is convergent with order1.0.Second,we improved a type of 3-stage stochastic Runge-Kutta method of order 1.0and constructed a class of balanced explicit 3-stage stochastic Runge-Kutta method.In the mean square sense,we also proved that it has the same convergence order as the original numerical method.Finally,for each improved stochastic Runge-Kutta method,we conducted numerical experiment to verify the correctness of the theoretical results.