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Simulation of non-lipschitz stochastic differential 2 equations driven by α-stable noise : a method based on deterministic homogenisation
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Gottwald, Georg A. and Melbourne, Ian (2021) Simulation of non-lipschitz stochastic differential 2 equations driven by α-stable noise : a method based on deterministic homogenisation. Multiscale Modeling and Simulation : A SIAM Interdisciplinary Journal, 19 (2). pp. 665-687. doi:10.1137/20M1333183
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Official URL: https://doi.org/10.1137/20M1333183
Abstract
We devise an explicit method to integrate α-stable stochastic differential equations (SDEs) with non-Lipschitz coefficients. To mitigate against numerical instabilities caused by unbounded increments of the Lévy noise, we use a deterministic map which has the desired SDE as its homogenised limit. Moreover, our method naturally overcomes difficulties in expressing the Marcus integral explicitly. We present an example of an SDE with a natural boundary showing that our method respects the boundary whereas Euler-Maruyama discretisation fails to do so. As a by-product we devise an entirely deterministic method to construct α-stable laws.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Lévy processes , Homogenization (Differential equations) , Multiscale modeling , Stochastic differential equations | ||||||||
| Journal or Publication Title: | Multiscale Modeling and Simulation : A SIAM Interdisciplinary Journal | ||||||||
| Publisher: | SIAM | ||||||||
| ISSN: | 1540-3459 | ||||||||
| Official Date: | 2021 | ||||||||
| Dates: |
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| Volume: | 19 | ||||||||
| Number: | 2 | ||||||||
| Page Range: | pp. 665-687 | ||||||||
| DOI: | 10.1137/20M1333183 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Publisher Statement: | “First Published in Multiscale Modeling and Simulation : A SIAM Interdisciplinary Journal in 19(2), published by the Society for Industrial and Applied Mathematics (SIAM)” | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Copyright Holders: | © by SIAM. Unauthorized reproduction of this article is prohibited | ||||||||
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