High Order Numerical Scheme For Hybrid Stochastic Differential Equations

Stochastic differential equations with Markovian switching as an important mathe-matical model are applied widely in many scientific fields,for example,finance,biology,demography and control etc.As the explicit solutions can be hardly obtained,con-structing appropriate numerical methods and discussing the properties of the numerical solutions are very essential in application and theory.In this paper,we consider the convergence and stability in the square sense of Milstein method.Applying generalized Ito’s formula and Markov property,we prove that the order of convergence of the method is one.The numerical example visu-ally demonstrates that the Milstein method can considerably improve the accuracy of simulation.For the stability in mean square,the conditons of stability and stepsize region are obtained.We give numerical experiments to support the conclusion which also reflect the influences of the step-size on stability of the numerical methods.In this paper,our study on the Milstein scheme provides a theoretical foundation for high order numerical method of hybrid stochastic differential equations.