| Due to the development of financial engineering,researching on numerical methods of stochastic differential equations(SDEs) draws much concern recently.The numerical stability is a very important characteristic of numerical methods,since an unstable numerical method may result in malignant growth of rounding error and distortion of the numerical solution,therefore studying on the numerical stability of SDEs is a very important issue.In this paper,almost sure exponential stability conditions of the Euler-Maruyama method for linear scalar SDEs is proved,and the stability regions of numerical solution for linear scalar SDEs is identified;By comparing with the mean-square stability regions reported by Saito and Mitsui(1996),the almost sure stability regions that is studied in this paper is larger than the mean-square stability regions,and therefore it is more valuable.Then in this paper,the backward Euler method and the stochastic theta method for numerical solution of stochastic differential equation is studied in depth.The stability of the backward Euler method and the stochastic theta method for numerical solution of the linear scalar SDE is proved.Moreover,in the linear growth condition and the one-sided Lipschitz condition,the stability of the stochastic theta method for the multi-dimensional non-linear stochastic differential equations is proved.Many significant conclusions have been made in numerical method for stochastic differential equations,but the Milstein method for stochastic differential equations have been rarely studied.In this paper,almost sure exponential stability conditions and p-th moment exponential stability conditions of the Milstein method for linear scalar SDEs is proved.The Milstein method is a first-order difference approximation model,while the Euler method is a half-order difference approximation model,therefore the Milstein method is better than the Euler method in convergence,and the stability of the Milstein method is more valuable in engineering.
|
PDF Full Text Request