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# Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis

CiteScore 2018: 0.66

SCImago Journal Rank (SJR) 2018: 0.319
Source Normalized Impact per Paper (SNIP) 2018: 0.720

Mathematical Citation Quotient (MCQ) 2018: 0.18

Online
ISSN
1569-3961
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Volume 20, Issue 1

# Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Håkon Hoel
/ Erik von Schwerin
/ Anders Szepessy
/ Raúl Tempone
Published Online: 2013-11-13 | DOI: https://doi.org/10.1515/mcma-2013-0014

## Abstract.

We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [Oper. Res. 56 (2008), 607–617] 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 $𝒪\left({\mathrm{TOL}}^{-3}\right)$ to $𝒪\left({\mathrm{TOL}}^{-2}log{\left({\mathrm{TOL}}^{-1}\right)}^{2}\right)$ for a mean square error of $𝒪\left({\mathrm{TOL}}^{2}\right)$. Later, the work [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217–234] presented an MLMC method using a hierarchy of adaptively refined, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretization MLMC method. This work improves the adaptive MLMC algorithms presented in [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217–234] 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 diffusion, where the complexity of a uniform single level method is $𝒪\left({\mathrm{TOL}}^{-4}\right)$. For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy $𝒪\left(\mathrm{TOL}\right)$ from $𝒪\left({\mathrm{TOL}}^{-3}\right)$ for the adaptive single level algorithm to essentially $𝒪\left({\mathrm{TOL}}^{-2}log{\left({\mathrm{TOL}}^{-1}\right)}^{2}\right)$ for the adaptive MLMC algorithm.

MSC: 65C30; 65Y20; 65L50; 65H35; 60H35; 60H10

Accepted: 2013-09-12

Published Online: 2013-11-13

Published in Print: 2014-03-01

Funding Source: Royal Institute of Technology in Stockholm

Award identifier / Grant number: Dahlquist fellowship

Funding Source: University of Austin Subcontract

Award identifier / Grant number: 024550

Funding Source: VR project

Award identifier / Grant number: “Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar”

Funding Source: King Abdullah University of Science and Technology (KAUST)

Citation Information: Monte Carlo Methods and Applications, Volume 20, Issue 1, Pages 1–41, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629,

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© 2014 by Walter de Gruyter Berlin/Boston.

## Citing Articles

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H. Hoel, K. H. Karlsen, N. H. Risebro, and E. B. Storrøsten
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