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Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

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Volume 23, Issue 1

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MCMC design-based non-parametric regression for rare event. Application to nested risk computations

Gersende Fort / Emmanuel Gobet
  • Corresponding author
  • Centre de Mathématiques Appliquées (CMAP), Ecole Polytechnique and CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France
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/ Eric Moulines
  • Centre de Mathématiques Appliquées (CMAP), Ecole Polytechnique and CNRS, Université Paris-Saclay,Route de Saclay, 91128 Palaiseau Cedex, France
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Published Online: 2017-02-03 | DOI: https://doi.org/10.1515/mcma-2017-0101

Abstract

We design and analyze an algorithm for estimating the mean of a function of a conditional expectation when the outer expectation is related to a rare event. The outer expectation is evaluated through the average along the path of an ergodic Markov chain generated by a Markov chain Monte Carlo sampler. The inner conditional expectation is computed as a non-parametric regression, using a least-squares method with a general function basis and a design given by the sampled Markov chain. We establish non-asymptotic bounds for the L2-empirical risks associated to this least-squares regression; this generalizes the error bounds usually obtained in the case of i.i.d. observations. Global error bounds are also derived for the nested expectation problem. Numerical results in the context of financial risk computations illustrate the performance of the algorithms.

Keywords: Empirical regression scheme; MCMC sampler; rare event

MSC 2010: 65C40; 62G08; 37M25

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About the article

Received: 2016-11-09

Accepted: 2017-01-19

Published Online: 2017-02-03

Published in Print: 2017-03-01


Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR-15-CE05-0024

The second author’s research is part of the Chair Financial Risks of the Risk Foundation, the Finance for Energy Market Research Centre and the ANR project CAESARS (ANR-15-CE05-0024).


Citation Information: Monte Carlo Methods and Applications, Volume 23, Issue 1, Pages 21–42, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma-2017-0101.

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