Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis

CiteScore 2017: 0.67

SCImago Journal Rank (SJR) 2017: 0.417
Source Normalized Impact per Paper (SNIP) 2017: 0.860

Mathematical Citation Quotient (MCQ) 2017: 0.25

See all formats and pricing
More options …
Volume 23, Issue 1


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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Eric Moulines
  • Centre de Mathématiques Appliquées (CMAP), Ecole Polytechnique and CNRS, Université Paris-Saclay,Route de Saclay, 91128 Palaiseau Cedex, France
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-02-03 | DOI: https://doi.org/10.1515/mcma-2017-0101


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


  • [1]

    Baraud Y., Comte F. and Viennet G., Adaptive estimation in autoregression or β-mixing regression via model selection, Ann. Statist. 29 (2001), no. 3, 839–875. Google Scholar

  • [2]

    Belomestny D., Kolodko A. and Schoenmakers J., Regression methods for stochastic control problems and their convergence analysis, SIAM J. Control Optim. 48 (2010), no. 5, 3562–3588. Google Scholar

  • [3]

    Blanchet J. and Lam H., State-dependent importance sampling for rare event simulation: An overview and recent advances, Surv. Oper. Res. Manag. Sci. 17 (2012), 38–59. Google Scholar

  • [4]

    Broadie M., Du Y. and Moallemi C. C., Risk Estimation via regression, Oper. Res. 63 (2015), no. 5, 1077–1097. Google Scholar

  • [5]

    Delattre S. and Gaïffas S., Nonparametric regression with martingale increment errors, Stochastic Process. Appl. 121 (2011), 2899–2924. Google Scholar

  • [6]

    Devineau L. and Loisel S., Construction d’un algorithme d’accélération de la méthode des “simulations dans les simulations” pour le calcul du capital économique solvabilité ii, Bull. Français d’Actuariat 10 (2009), no. 17, 188–221. Google Scholar

  • [7]

    Douc R., Fort G., Moulines E. and Soulier P., Practical drift conditions for subgeometric rates of convergence, Ann. Appl. Probab. 14 (2004), no. 3, 1353–1377. Google Scholar

  • [8]

    Egloff D., Monte Carlo algorithms for optimal stopping and statistical learning, Ann. Appl. Probab. 15 (2005), 1396–1432. Google Scholar

  • [9]

    Fort G. and Moulines E., Convergence of the Monte Carlo expectation maximization for curved exponential families, Ann. Statist. 31 (2003), no. 4, 1220–1259. Google Scholar

  • [10]

    Fort G. and Moulines E., Polynomial ergodicity of Markov transition kernels, Stochastic Process. Appl. 103 (2003), no. 1, 57–99. Google Scholar

  • [11]

    Gobet E. and Liu G., Rare event simulation using reversible shaking transformations, SIAM J. Sci. Comput. 37 (2015), no. 5, A2295–A2316. Google Scholar

  • [12]

    Gobet E. and Turkedjiev P., Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions, Math. Comp. 299 (2016), no. 85, 1359–1391. Google Scholar

  • [13]

    Gordy M. B. and Juneja S., Nested simulation in portfolio risk measurement, Manag. Sci. 56 (2010), no. 10, 1833–1848. Google Scholar

  • [14]

    Gyorfi L., Kohler M., Krzyzak A. and Walk H., A Distribution-Free Theory of Nonparametric Regression, Springer Ser. Statist., Springer, New York, 2002. Google Scholar

  • [15]

    Hong L. J. and Juneja S., Estimating the mean of a non-linear function of conditional expectation, Proceedings of the 2009 Winter Simulation Conference (WSC), IEEE Press, Piscataway (2009), 1223–1236. Google Scholar

  • [16]

    Lemor J-P., Gobet E. and Warin X., Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations, Bernoulli 12 (2006), no. 5, 889–916. Google Scholar

  • [17]

    Liu M. and Staum J., Stochastic kriging for efficient nested simulation of expected shortfall, J. Risk 12 (2010), no. 3, 3–27. Google Scholar

  • [18]

    Longstaff F. and Schwartz E. S., Valuing American options by simulation: A simple least squares approach, Rev. Financ. Stud. 14 (2001), 113–147. Google Scholar

  • [19]

    McNeil A. J., Frey R. and Embrechts P., Quantitative Risk Management, Princeton Ser. Finance, Princeton University Press, Princeton, 2005. Google Scholar

  • [20]

    Meyn S. P. and Tweedie R. L., Markov Chains and Stochastic Stability, Springer, Berlin, 1993. Google Scholar

  • [21]

    Ren Q. and Mojirsheibani M., A note on nonparametric regression with β-mixing sequences, Comm. Statist. Theory Methods 39 (2010), no. 12, 2280–2287. Google Scholar

  • [22]

    Rosenthal J. S., Optimal Proposal Distributions and Adaptive MCMC, Chapman & Hall/CRC Handb. Mod. Stat. Methods, CRC Press, Boca Raton, 2008. Google Scholar

  • [23]

    Rubinstein R. Y. and Kroese D. P., Simulation and the Monte-Carlo Method, 2nd ed., Wiley Ser. Probab. Stat., John Wiley & Sons, Hoboken, 2008. Google Scholar

  • [24]

    Tsitsiklis J. N. and Van Roy B., Regression methods for pricing complex American-style options, IEEE Trans. Neural Netw. 12 (2001), no. 4, 694–703. Google Scholar

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.

Export Citation

© 2017 by De Gruyter.Get Permission

Comments (0)

Please log in or register to comment.
Log in