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Adaptive Multilevel Monte Carlo Simulation

  • Håkon Hoel
  • Erik von SchwerinEmail author
  • Anders Szepessy
  • Raúl Tempone
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 82)

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 \( \vartheta{\rm(TOL),\, from\,(TOL^{-3})}\), from using a single level version of the adaptive algorithm to \( \vartheta\left( \begin{array}{lll}\left({(TOL^{-1})\,log(TOL)}\right)^2\end{array}\right).\)

Keywords

Stochastic Differential Equation Adaptive Algorithm Single Level Monte Carlo Algorithm Error Indicator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Håkon Hoel
    • 2
  • Erik von Schwerin
    • 1
    Email author
  • Anders Szepessy
    • 3
  • Raúl Tempone
    • 1
  1. 1.Applied Mathematics and Computational SciencesKAUSTThuwalSaudi Arabia
  2. 2.Department of Numerical AnalysisCSC, Royal Institute of Technology (KTH)StockholmSweden
  3. 3.Department of MathematicsRoyal Institute of Technology (KTH)StockholmSweden

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