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Almost sure and moment exponential stability of Milstein methods forstochastic differential equations
ZHANG Yu-Xin 1,WANG Peng 2 *
1.College of Science, Harbin Engineering University, Harbin 150001
2.College of Mathematics, Jilin University, Changchun 130012
*Correspondence author
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Funding: Specialized Research Fund for the Doctoral Program of Higher Education （No.20090061120038）
Opened online:29 January 2013
Accepted by: none
Citation: ZHANG Yu-Xin,WANG Peng.Almost sure and moment exponential stability of Milstein methods forstochastic differential equations[OL]. [29 January 2013] http://en.paper.edu.cn/en_releasepaper/content/4516844
 
 
In this paper, almost sure and moment exponential stability of explicit and semi-implicit Milstein methods for stochastic differential equations (SDEs) is studied. Motivated by the work of Higham, Mao and Yuan [Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations], we investigate stability properties in the limit as the timestep tends to zero. The analysis proceeds as follows: We begin by showing that when scalar-noise SDE obeys some proper conditions, the Milstein method reproduces almost sure and small moment exponential stability for sufficiently small stepsizes. Then similar conclusions for the semi-implicit Milstein scheme is proved when the drift coefficient of the SDE obeys a one-sided Lipschitz condition. Finally We generalize our results to multidimensional noise SDEs.
Keywords:Milstein scheme; one-sided Lipschitz condition; exponential stability; linear growth condition
 
 
 

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