Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Handle URI:
http://hdl.handle.net/10754/563323
Title:
Implementation and analysis of an adaptive multilevel Monte Carlo algorithm
Authors:
Hoel, Hakon; Von Schwerin, Erik; Szepessy, Anders; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic dierential equations (SDE). The work [11] proposed and analyzed an MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a single level Euler-Maruyama Monte Carlo method from O(TOL-3) to O(TOL-2 log(TOL-1)2) for a mean square error of O(TOL2). Later, the work [17] presented an MLMC method using a hierarchy of adaptively re ned, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretizationMLMC method. This work improves the adaptiveMLMC algorithms presented in [17] and it also provides mathematical analysis of the improved algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diusion, where the complexity of a uniform single level method is O(TOL-4). For both these cases the results con rm the theory, exhibiting savings in the computational cost for achieving the accuracy O(TOL) from O(TOL-3) for the adaptive single level algorithm to essentially O(TOL-2 log(TOL-1)2) for the adaptive MLMC algorithm. © 2014 by Walter de Gruyter Berlin/Boston 2014.
KAUST Department:
Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Stochastic Numerics Research Group
Publisher:
Walter de Gruyter GmbH
Journal:
Monte Carlo Methods and Applications
Issue Date:
1-Jan-2014
DOI:
10.1515/mcma-2013-0014
Type:
Article
ISSN:
09299629
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorHoel, Hakonen
dc.contributor.authorVon Schwerin, Eriken
dc.contributor.authorSzepessy, Andersen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-08-03T11:45:44Zen
dc.date.available2015-08-03T11:45:44Zen
dc.date.issued2014-01-01en
dc.identifier.issn09299629en
dc.identifier.doi10.1515/mcma-2013-0014en
dc.identifier.urihttp://hdl.handle.net/10754/563323en
dc.description.abstractWe present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic dierential equations (SDE). The work [11] proposed and analyzed an MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a single level Euler-Maruyama Monte Carlo method from O(TOL-3) to O(TOL-2 log(TOL-1)2) for a mean square error of O(TOL2). Later, the work [17] presented an MLMC method using a hierarchy of adaptively re ned, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretizationMLMC method. This work improves the adaptiveMLMC algorithms presented in [17] and it also provides mathematical analysis of the improved algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diusion, where the complexity of a uniform single level method is O(TOL-4). For both these cases the results con rm the theory, exhibiting savings in the computational cost for achieving the accuracy O(TOL) from O(TOL-3) for the adaptive single level algorithm to essentially O(TOL-2 log(TOL-1)2) for the adaptive MLMC algorithm. © 2014 by Walter de Gruyter Berlin/Boston 2014.en
dc.publisherWalter de Gruyter GmbHen
dc.subjecta posteriori error estimatesen
dc.subjectadaptivityen
dc.subjectadjointsen
dc.subjectbackward dual functionsen
dc.subjectComputational financeen
dc.subjecterror controlen
dc.subjectEuler-Maruyama methoden
dc.subjectMonte Carloen
dc.subjectmultilevelen
dc.subjectweak approximationen
dc.titleImplementation and analysis of an adaptive multilevel Monte Carlo algorithmen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentStochastic Numerics Research Groupen
dc.identifier.journalMonte Carlo Methods and Applicationsen
dc.contributor.institutionCSQI-MATHICSE, École Polytechnique Fédérale de Lausanne, Switzerlanden
dc.contributor.institutionDepartment of Mathematics, Royal Institute of Technology (KTH), Stockholm, Swedenen
kaust.authorHoel, Hakonen
kaust.authorTempone, Raulen
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