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

Author

Listed:
  • Fort Gersende

    (LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay, 75013, Paris, France)

  • Gobet Emmanuel

    (Centre de Mathématiques Appliquées (CMAP), Ecole Polytechnique and CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France)

  • Moulines Eric

    (Centre de Mathématiques Appliquées (CMAP), Ecole Polytechnique and CNRS, Université Paris-Saclay,Route de Saclay, 91128 Palaiseau Cedex, France)

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${L_{2}}$-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.

Suggested Citation

  • Fort Gersende & Gobet Emmanuel & Moulines Eric, 2017. "MCMC design-based non-parametric regression for rare event. Application to nested risk computations," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 21-42, March.
    • Handle: RePEc:bpj:mcmeap:v:23:y:2017:i:1:p:21-42:n:3
    DOI: 10.1515/mcma-2017-0101
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    References listed on IDEAS

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    5. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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