Mean-square stability of two classes of theta Milstein methods for nonlinear stochastic differential equations
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Recently, there is growing interest in developing new numerical methods for stochastic di erential equations (SDEs), in order to improve the stability of approximation solution. There are many numerical methods have been constructed based on a Milstein scheme for SDEs. However, there exists very little results on the stability analysis of Milstein type methods for SDEs. This paper is concerned with mean-square (MS) stability of the semi-implicit theta Milstein methods and drifting split-step theta Milstein methods for nonlinear stochastic di erential equations. Under a coupled condition on the drifting and di usion coecients, it is proved that, the methods with > 1 2 are unconditionally preserve the MS-stability of the SDEs. For 2 [0; 1 2 ], the methods are MS-stable for some small step-size. This work is di erent from the previous works such that we could get rid of the restrictions that existed on the step-size of two classes Milstein methods for a symptomatic mean-square stability of the non-linear stochastic di erential equations, under Local lipschitz condition. Numerical experiments are given to demonstrate the conclusions.
Abstract
Recently, there is growing interest in developing new numerical methods for stochastic di erential equations (SDEs), in order to improve the stability of approximation solution. There are many numerical methods have been constructed based on a Milstein scheme for SDEs. However, there exists very little results on the stability analysis of Milstein type methods for SDEs. This paper is concerned with mean-square (MS) stability of the semi-implicit theta Milstein methods and drifting split-step theta Milstein methods for nonlinear stochastic di erential equations. Under a coupled condition on the drifting and di usion coecients, it is proved that, the methods with > 1 2 are unconditionally preserve the MS-stability of the SDEs. For 2 [0; 1 2 ], the methods are MS-stable for some small step-size. This work is di erent from the previous works such that we could get rid of the restrictions that existed on the step-size of two classes Milstein methods for a symptomatic mean-square stability of the non-linear stochastic di erential equations, under Local lipschitz condition. Numerical experiments are given to demonstrate the conclusions.
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