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Strong convergence of Euler-type methods for nonlinear stochastic differential equations
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Higham, Desmond J., Mao, Xuerong and Stuart, A. M. (2002) Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM Journal on Numerical Analysis, 40 (3). pp. 1041-1063. doi:10.1137/S0036142901389530 ISSN 0036-1429.
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Official URL: http://dx.doi.org/10.1137/S0036142901389530
Abstract
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.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | SIAM Journal on Numerical Analysis | ||||
Publisher: | SIAM Publications | ||||
ISSN: | 0036-1429 | ||||
Official Date: | 12 September 2002 | ||||
Dates: |
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Volume: | 40 | ||||
Number: | 3 | ||||
Number of Pages: | 23 | ||||
Page Range: | pp. 1041-1063 | ||||
DOI: | 10.1137/S0036142901389530 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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