Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-31T07:28:39.689Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Acceleration of the Multi-Level Monte Carlo Method

Part of: Stochastic analysis Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 January 2018

Kristian Debrabant  and
Andreas Röβler
Kristian Debrabant*
Affiliation:
University of Southern Denmark
Andreas Röβler*
Affiliation:
Universität zu Lübeck
*
∗ Postal address: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark.
∗∗ Postal address: Institut für Mathematik, Universität zu Lübeck, Ratzeburger Allee 160, D-23562 Lübeck, Germany. Email address: roessler@math.uni-luebeck.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The multi-level Monte Carlo method proposed by Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor (p / α)2 if weak approximation methods of orders α and p are applied in the case of computational costs growing with the same order as the variances decay.

Keywords

Multi-level Monte CarloMonte Carloweak approximationvariance reductionstochastic differential equation

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60H35: Computational methods for stochastic equations 65C20: Models, numerical methods 68U20: Simulation
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 307 - 322
Copyright
© Applied Probability Trust 

References

Avikainen, R. (2009). {{On irregular functionals of SDEs and the Euler scheme}}. Finance Stoch. 13, 381–401.Google Scholar
Debrabant, K. (2010). {{Runge–Kutta methods for third order weak approximation of SDEs with multidimensional additive noise}}. BIT 50, 541–558.Google Scholar
Dereich, S. (2011). {{Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction}}. Ann. Appl. Prob. 21, 283–311.Google Scholar
Duffie, D. and Glynn, P. (1995). {{Efficient Monte Carlo simulation of security prices}}. Ann. Appl. Prob. 5, 897–905.Google Scholar
Giles, M. (2008). {{Improved multilevel Monte Carlo convergence using the Milstein scheme}}. In Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, Berlin, pp. 343–358.Google Scholar
Giles, M. B. (2008). {{Multilevel Monte Carlo path simulation}}. Operat. Res. 56, 607–617.Google Scholar
Giles, M. B., Higham, D. J. and Mao, X. (2009). {{Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff}}. Finance Stoch. 13, 403–413.Google Scholar
Heinrich, S. (2001). {Multilevel Monte Carlo methods}. In Large-Scale Scientific Computing (Lecture Notes Comput. Sci. 2179), Springer, Berlin, pp. 58–67.Google Scholar
Kebaier, A. (2005). {{Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing}}. Ann. Appl. Prob. 15, 2681–2705.Google Scholar
Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations (Appl. Math. (New York) 23). Springer, Berlin.Google Scholar
Kloeden, P. E., Neuenkirch, A. and Pavani, R. (2011). {{Multilevel Monte Carlo for stochastic differential equations with additive fractional noise}}. Ann. Operat. Res. 189, 255–276.Google Scholar
Rössler, A. (2009). {{Second order Runge–Kutta methods for Itô stochastic differential equations}}. SIAM J. Numerical Anal. 47, 1713–1738.Google Scholar
Schwab, C. and Gittelson, C. J. (2011). {{Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs}}. Acta Numerica 20, 291–467.Google Scholar