Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

Handle URI:
http://hdl.handle.net/10754/622138
Title:
Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo
Authors:
Hoel, Hakon; Häppölä, Juho; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
A formal mean square error expansion (MSE) is derived for Euler-Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise, a posteriori, adaptive time-stepping Euler-Maruyama algorithm for numerical solutions of SDE, and the resulting algorithm is incorporated into a multilevel Monte Carlo (MLMC) algorithm for weak approximations of SDE. This gives an efficient MSE adaptive MLMC algorithm for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC algorithm is shown to outperform the uniform time-stepping MLMC algorithm by orders of magnitude, producing output whose error with high probability is bounded by TOL > 0 at the near-optimal MLMC cost rate б(TOL log(TOL)) that is achieved when the cost of sample generation is б(1).
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Hoel H, Häppölä J, Tempone R (2016) Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo. Monte Carlo and Quasi-Monte Carlo Methods: 29–86. Available: http://dx.doi.org/10.1007/978-3-319-33507-0_2.
Publisher:
Springer Nature
Journal:
Springer Proceedings in Mathematics & Statistics
Conference/Event name:
11th International Conference on Monte Carlo and Quasi Monte Carlo Methods in Scientific Computing, MCQMC 2014
Issue Date:
13-Jun-2016
DOI:
10.1007/978-3-319-33507-0_2
Type:
Conference Paper
ISSN:
2194-1009; 2194-1017
Appears in Collections:
Conference Papers; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorHoel, Hakonen
dc.contributor.authorHäppölä, Juhoen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2017-01-02T08:10:20Z-
dc.date.available2017-01-02T08:10:20Z-
dc.date.issued2016-06-13en
dc.identifier.citationHoel H, Häppölä J, Tempone R (2016) Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo. Monte Carlo and Quasi-Monte Carlo Methods: 29–86. Available: http://dx.doi.org/10.1007/978-3-319-33507-0_2.en
dc.identifier.issn2194-1009en
dc.identifier.issn2194-1017en
dc.identifier.doi10.1007/978-3-319-33507-0_2en
dc.identifier.urihttp://hdl.handle.net/10754/622138-
dc.description.abstractA formal mean square error expansion (MSE) is derived for Euler-Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise, a posteriori, adaptive time-stepping Euler-Maruyama algorithm for numerical solutions of SDE, and the resulting algorithm is incorporated into a multilevel Monte Carlo (MLMC) algorithm for weak approximations of SDE. This gives an efficient MSE adaptive MLMC algorithm for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC algorithm is shown to outperform the uniform time-stepping MLMC algorithm by orders of magnitude, producing output whose error with high probability is bounded by TOL > 0 at the near-optimal MLMC cost rate б(TOL log(TOL)) that is achieved when the cost of sample generation is б(1).en
dc.publisherSpringer Natureen
dc.subjectA posteriori error estimationen
dc.subjectAdaptive methodsen
dc.subjectAdjointsen
dc.subjectEuler-maruyama methoden
dc.subjectMultilevel monte carloen
dc.subjectStochastic differential equationsen
dc.titleConstruction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carloen
dc.typeConference Paperen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalSpringer Proceedings in Mathematics & Statisticsen
dc.conference.date2014-04-06 to 2014-04-11en
dc.conference.name11th International Conference on Monte Carlo and Quasi Monte Carlo Methods in Scientific Computing, MCQMC 2014en
dc.conference.locationLeuven, BELen
dc.contributor.institutionDepartment of Mathematics, University of Oslo, P.O. Box 1053, Blindern, Oslo, Norwayen
kaust.authorHoel, Hakonen
kaust.authorHäppölä, Juhoen
kaust.authorTempone, Raulen
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