Milstein Method For Two Types Of Stochastic Differential Equations Based On Double Integral Approximation

Stochastic differential equations(SDEs) play a significant role in mathematical modeling of random phenomena which can not be replaced by traditional deterministic models. However, in many stochastic problems, it is quite difficult and complex to compute double stochastic integrals which generated by two independent Brown motions.In this paper, we come up with a new idea to approximate the double stochastic integrals by using cumulative sum. A new split-step Milstein method is presented based on the new approximation for SDEs with multi-dimensional noise. Then a new Milstein method is provided for SDEs with constant delay by using the similar approximation idea.Afterwards we analyze the strong order of convergence and investigate the mean-square stability properties for two different kinds of SDEs. And we achieve some results: The new Milstein methods for two kinds of SDEs still keep the strong convergence order 1.0 under certain conditions; For SDEs with multi-dimensional noise, the sufficient and necessary condition of mean-square stability is achieved under multi-dimentional coefficient case. And the sufficient condition of mean-square stability is presented under one-dimentional case; For SDEs with constand delay, we have the sufficient condition of mean-square stability under different coefficients and step size.Numerical examples are provided to verify all the conclusions mentioned before and demonstrate the effectiveness and reliability of the new numerical methods.