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A framework for adaptive Monte-Carlo procedures

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  • Bernard Lapeyre

    (CERMICS)

  • J'er^ome Lelong

    (LJK)

Abstract

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Vazquez-Abad and Dufresne, Fu and Su, and Arouna. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives.

Suggested Citation

  • Bernard Lapeyre & J'er^ome Lelong, 2010. "A framework for adaptive Monte-Carlo procedures," Papers 1001.3551, arXiv.org, revised Jul 2010.
    • Handle: RePEc:arx:papers:1001.3551
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    References listed on IDEAS

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    1. Paul Glasserman & Philip Heidelberger & Perwez Shahabuddin, 1999. "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path‐Dependent Options," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 117-152, April.
    2. Kawai Reiichiro, 2006. "An importance sampling method based on the density transformation of Lévy processes," Monte Carlo Methods and Applications, De Gruyter, vol. 12(2), pages 171-186, April.
    3. Arouna Bouhari, 2004. "Adaptative Monte Carlo Method, A Variance Reduction Technique," Monte Carlo Methods and Applications, De Gruyter, vol. 10(1), pages 1-24, March.
    4. Sujin Kim & Shane G. Henderson, 2007. "Adaptive Control Variates for Finite-Horizon Simulation," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 508-527, August.
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