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Aufsatz / Paper in SCI-Expanded-Zeitschrift

An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and mean-square unstable equilibrium

Buckwar E., Ableidinger M., Thalhammer A.: An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and mean-square unstable equilibrium, in: Journal of Computational and Applied Mathematics, Volume 316, 2017.

BibTeX

@ARTICLE{
title = {An importance sampling technique in Monte Carlo methods for SDEs with a.s. stable and mean-square unstable equilibrium},
type = {Aufsatz / Paper in SCI-Expanded-Zeitschrift},
author = {Buckwar, Evelyn and Ableidinger, Markus and Thalhammer, Andreas},
language = {EN},
abstract = {In this work we investigate the interplay of almost sure and mean-square stability for linear SDEs and the Monte Carlo method for estimating the second moment of the solution process. In the situation where the zero solution of the SDE is asymptotically stable in the almost sure sense but asymptotically mean-square unstable, the latter property is determined by rarely occurring trajectories that are sufficiently far away from the origin. The standard Monte Carlo approach for estimating higher moments essentially computes a finite number of trajectories and is bound to miss those rare events. It thus fails to reproduce the correct mean-square dynamics (under reasonable cost). A straightforward application of variance reduction techniques will typically not resolve the situation unless these methods force the rare, exploding trajectories to happen more frequently. Here we propose an appropriately tuned importance sampling technique based on Girsanov’s theorem to deal with the rare event simulation. In addition further variance reduction techniques, such as multilevel Monte Carlo, can be applied to control the variance of the modified Monte Carlo estimators. As an illustrative example we discuss the numerical treatment of the stochastic heat equation with multiplicative noise and present simulation results.},
publisher = {Elsevier B.V.},
journal = {Journal of Computational and Applied Mathematics},
volume = {316},
issn = {0377-0427},
year = {2017},
}

Details

Zusammenfassung: In this work we investigate the interplay of almost sure and mean-square stability for linear SDEs and the Monte Carlo method for estimating the second moment of the solution process. In the situation where the zero solution of the SDE is asymptotically stable in the almost sure sense but asymptotically mean-square unstable, the latter property is determined by rarely occurring trajectories that are sufficiently far away from the origin. The standard Monte Carlo approach for estimating higher moments essentially computes a finite number of trajectories and is bound to miss those rare events. It thus fails to reproduce the correct mean-square dynamics (under reasonable cost). A straightforward application of variance reduction techniques will typically not resolve the situation unless these methods force the rare, exploding trajectories to happen more frequently. Here we propose an appropriately tuned importance sampling technique based on Girsanov’s theorem to deal with the rare event simulation. In addition further variance reduction techniques, such as multilevel Monte Carlo, can be applied to control the variance of the modified Monte Carlo estimators. As an illustrative example we discuss the numerical treatment of the stochastic heat equation with multiplicative noise and present simulation results.

Journal: Journal of Computational and Applied Mathematics
Volume: 316
Erscheinungsjahr: 2017
Anzahl Seiten: 12
DOI: http://dx.doi.org/10.1016/j.cam.2016.08.043
Verlag: Elsevier B.V.
ISSN: 0377-0427
Reichweite: International

Beteiligte

AutorInnen / HerausgeberInnen: Univ.-Prof. Dr. Evelyn Buckwar, DI Markus Ableidinger, DI Andreas Thalhammer

Forschungseinheiten der JKU:

Wissenschaftszweige: 101 Mathematik | 101014 Numerische Mathematik | 101018 Statistik | 101019 Stochastik | 101024 Wahrscheinlichkeitstheorie

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