Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

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• 作者：Adam Andersson ; Raphael Kruse
• 关键词：SODE ; Backward Euler–Maruyama method ; BDF2 ; Maruyama method ; Strong convergence rates ; Global monotoncity condition
• 刊名：BIT Numerical Mathematics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：57
• 期：1
• 页码：21-53
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Computational Mathematics and Numerical Analysis; Numeric Computing; Mathematics, general;
• 出版者：Springer Netherlands
• ISSN：1572-9125
• 卷排序：57

文摘

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be \(\frac{1}{2}\) for the two-step BDF-Maruyama scheme and for the backward Euler–Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the \(\tfrac{3}{2}\)-volatility model from finance and a two dimensional problem related to Galerkin approximation of SPDE, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity.