Adaptive Multilevel Monte Carlo Simulation

Handle URI:
http://hdl.handle.net/10754/575782
Title:
Adaptive Multilevel Monte Carlo Simulation
Authors:
Hoel, H; von Schwerin, E; Szepessy, A; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
This work generalizes a multilevel forward Euler Monte Carlo method introduced in Michael B. Giles. (Michael Giles. Oper. Res. 56(3):607–617, 2008.) for the approximation of expected values depending on the solution to an Itô stochastic differential equation. The work (Michael Giles. Oper. Res. 56(3):607– 617, 2008.) proposed and analyzed a forward Euler multilevelMonte Carlo method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretizations, generated by an adaptive algorithmintroduced in (AnnaDzougoutov et al. Raùl Tempone. Adaptive Monte Carlo algorithms for stopped diffusion. In Multiscale methods in science and engineering, volume 44 of Lect. Notes Comput. Sci. Eng., pages 59–88. Springer, Berlin, 2005; Kyoung-Sook Moon et al. Stoch. Anal. Appl. 23(3):511–558, 2005; Kyoung-Sook Moon et al. An adaptive algorithm for ordinary, stochastic and partial differential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325–343. Amer. Math. Soc., Providence, RI, 2005.). This form of the adaptive algorithm generates stochastic, path dependent, time steps and is based on a posteriori error expansions first developed in (Anders Szepessy et al. Comm. Pure Appl. Math. 54(10):1169– 1214, 2001). Our numerical results for a stopped diffusion problem, exhibit savings in the computational cost to achieve an accuracy of ϑ(TOL),from(TOL−3), from using a single level version of the adaptive algorithm to ϑ(((TOL−1)log(TOL))2).
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Applied Mathematics and Computational Science Program; Applied Mathematics and Computational Science Program; Stochastic Numerics Research Group
Publisher:
Springer Science + Business Media
Journal:
Lecture Notes in Computational Science and Engineering
Conference/Event name:
Proceedings of a Winter Workshop at the Banff International Research Station
Issue Date:
23-Aug-2011
DOI:
10.1007/978-3-642-21943-6_10
Type:
Conference Paper
ISSN:
1439-7358
ISBN:
978-3-642-21942-9
Appears in Collections:
Conference Papers; Applied Mathematics and Computational Science Program; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorHoel, Hen
dc.contributor.authorvon Schwerin, Een
dc.contributor.authorSzepessy, Aen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-08-24T09:26:04Zen
dc.date.available2015-08-24T09:26:04Zen
dc.date.issued2011-08-23en
dc.identifier.isbn978-3-642-21942-9en
dc.identifier.issn1439-7358en
dc.identifier.doi10.1007/978-3-642-21943-6_10en
dc.identifier.urihttp://hdl.handle.net/10754/575782en
dc.description.abstractThis work generalizes a multilevel forward Euler Monte Carlo method introduced in Michael B. Giles. (Michael Giles. Oper. Res. 56(3):607–617, 2008.) for the approximation of expected values depending on the solution to an Itô stochastic differential equation. The work (Michael Giles. Oper. Res. 56(3):607– 617, 2008.) proposed and analyzed a forward Euler multilevelMonte Carlo method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretizations, generated by an adaptive algorithmintroduced in (AnnaDzougoutov et al. Raùl Tempone. Adaptive Monte Carlo algorithms for stopped diffusion. In Multiscale methods in science and engineering, volume 44 of Lect. Notes Comput. Sci. Eng., pages 59–88. Springer, Berlin, 2005; Kyoung-Sook Moon et al. Stoch. Anal. Appl. 23(3):511–558, 2005; Kyoung-Sook Moon et al. An adaptive algorithm for ordinary, stochastic and partial differential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325–343. Amer. Math. Soc., Providence, RI, 2005.). This form of the adaptive algorithm generates stochastic, path dependent, time steps and is based on a posteriori error expansions first developed in (Anders Szepessy et al. Comm. Pure Appl. Math. 54(10):1169– 1214, 2001). Our numerical results for a stopped diffusion problem, exhibit savings in the computational cost to achieve an accuracy of ϑ(TOL),from(TOL−3), from using a single level version of the adaptive algorithm to ϑ(((TOL−1)log(TOL))2).en
dc.publisherSpringer Science + Business Mediaen
dc.titleAdaptive Multilevel Monte Carlo Simulationen
dc.typeConference Paperen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentStochastic Numerics Research Groupen
dc.identifier.journalLecture Notes in Computational Science and Engineeringen
dc.conference.nameProceedings of a Winter Workshop at the Banff International Research Stationen
kaust.authorvon Schwerin, Eriken
kaust.authorTempone, Raulen
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