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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

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CiteScore 2017: 0.67

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1569-3961
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Volume 16, Issue 1

Issues

Adaptive weak approximation of reflected and stopped diffusions

Christian Bayer
  • Institute for Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden, currently at Institute of Mathematics, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. E-mail:
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Anders Szepessy
  • Institute for Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden. E-mail:
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Raúl Tempone
Published Online: 2010-04-21 | DOI: https://doi.org/10.1515/mcma.2010.001

Abstract

We study the weak approximation problem of diffusions, which are reflected at a subset of the boundary of a domain and stopped at the remaining boundary. First, we derive an error representation for the projected Euler method of Costantini, Pacchiarotti and Sartoretto [Costantini et al., SIAM J. Appl. Math., 58(1):73–102, 1998], based on which we introduce two new algorithms. The first one uses a correction term from the representation in order to obtain a higher order of convergence, but the computation of the correction term is, in general, not feasible in dimensions d > 1. The second algorithm is adaptive in the sense of Moon, Szepessy, Tempone and Zouraris [Moon et al., Stoch. Anal. Appl., 23:511–558, 2005], using stochastic refinement of the time grid based on a computable error expansion derived from the representation. Regarding the stopped diffusion, it is based in the adaptive algorithm for purely stopped diffusions presented in Dzougoutov, Moon, von Schwerin, Szepessy and Tempone [Dzougoutov et al., Lect. Notes Comput. Sci. Eng., 44, 59–88, 2005]. We give numerical examples underlining the theoretical results.

About the article

Received: 2009-07-14

Revised: 2010-02-01

Published Online: 2010-04-21

Published in Print: 2010-04-01


Citation Information: Monte Carlo Methods and Applications, Volume 16, Issue 1, Pages 1–67, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma.2010.001.

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[3]
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Computational Optimization and Applications, 2015, Volume 61, Number 1, Page 101
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