| Issue |
ESAIM: PS
Volume 19, 2015
|
|
|---|---|---|
| Page(s) | 361 - 394 | |
| DOI | https://doi.org/10.1051/ps/2014029 | |
| Published online | 26 October 2015 | |
Analysis of adaptive multilevel splitting algorithms in an idealized case∗
1
UniversitéParis-Est, CERMICS (ENPC), INRIA, 6-8 avenue Blaise Pascal,
77455
Marne-la-Vallée,
France
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2
INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, B.P.
105, 78153
Le Chesnay,
France
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Received: 5 May 2014
Revised: 1 September 2014
Abstract
The Adaptive Multilevel Splitting algorithm [F. Cérou and A. Guyader, Stoch. Anal. Appl. 25 (2007) 417–443] is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the k less well-adapted particles among n are killed while k new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever k. We also obtain a precise asymptotic expansion for the variance of the estimator and the cost of the algorithm in the large n limit, for a fixed k.
Mathematics Subject Classification: 65C05 / 65C35 / 62G30
Key words: Monte-Carlo simulation / rare events / multilevel splitting
This work was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492.
© EDP Sciences, SMAI, 2015
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