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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. 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 |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Faculty of Science > Mathematics |
| Journal or Publication Title: | SIAM Journal on Numerical Analysis |
| Publisher: | SIAM Publications |
| ISSN: | 0036-1429 |
| Date: | 12 September 2002 |
| Volume: | 40 |
| Number: | 3 |
| Number of Pages: | 23 |
| Page Range: | pp. 1041-1063 |
| Identification Number: | 10.1137/S0036142901389530 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| URI: | http://wrap.warwick.ac.uk/id/eprint/10529 |
Data sourced from Thomson Reuters' Web of Knowledge
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