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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

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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
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > 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:
DateEvent
2021Published
15 April 2021Available
20 January 2021Accepted
Volume: 19
Number: 2
Page Range: pp. 665-687
DOI: 10.1137/20M1333183
Status: Peer Reviewed
Publication Status: Published
Reuse Statement (publisher, data, author rights): “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
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
AdG 320977European Research Councilhttp://dx.doi.org/10.13039/501100000781
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