Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

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• 作者：Wolf-Jürgen Beyn ; Elena Isaak ; Raphael Kruse
• 关键词：Stochastic differential equations ; Global monotonicity condition ; Split ; step backward Milstein method ; Projected Milstein method ; Mean ; square convergence ; Strong convergence ; C ; stability ; B ; consistency
• 刊名：Journal of Scientific Computing
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：70
• 期：3
• 页码：1042-1077
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algorithms; Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer US
• ISSN：1573-7691
• 卷排序：70

文摘

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in Beyn et al. (J Sci Comput 67(3):955–987, 2016. doi:10.1007/s10915-015-0114-4). As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler–Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.