Strong convergence of Euler-type methods for nonlinear stochastic differential equations

被引:506
作者
Higham, DJ [1 ]
Mao, XR
Stuart, AM
机构
[1] Univ Strathclyde, Dept Math, Glasgow G1 1XH, Lanark, Scotland
[2] Univ Strathclyde, Dept Stat & Modelling Sci, Glasgow G1 1XH, Lanark, Scotland
[3] Univ Warwick, Inst Math, Coventry CV4 7AL, W Midlands, England
关键词
backward Euler; Euler-Maruyama; finite-time convergence; implicit; moment bounds; nonlinearity; one-sided Lipschitz condition; split-step;
D O I
10.1137/S0036142901389530
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler-Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p > 2. As an application of this general theory we show that an implicit variant of Euler-Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler-Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.
引用
收藏
页码:1041 / 1063
页数:23
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