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Rare event simulation for diffusion processes via two-stage importance sampling

Author

Listed:
  • Metzler Adam

    (Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, Canada)

  • Scott Alexandre

    (Western University, 1151 Richmond Street North, London, Ontario, Canada)

Abstract

We consider the problem of estimating expected values of functionals of real-valued diffusions over regions in path space that have very small probability. We propose a two-stage importance sampling procedure that first converts the problem into one involving standard Brownian motion and then addresses the rare event problem in this simpler setting. In order to identify an effective yet practical importance measure we propose using a time-dependent deterministic drift that minimizes the relative entropy between the corresponding importance measure and the conditional law of the standard Brownian motion, given that its trajectory lies in the region of interest. We provide numerical evidence that (i) our entropy-based criteria performs favourably with an alternative, but less general and less practical, criteria based on large deviations and (ii) our two-stage procedure performs admirably in cases where the region of interest is so rare that crude estimators fail completely.

Suggested Citation

  • Metzler Adam & Scott Alexandre, 2014. "Rare event simulation for diffusion processes via two-stage importance sampling," Monte Carlo Methods and Applications, De Gruyter, vol. 20(2), pages 77-100, June.
    • Handle: RePEc:bpj:mcmeap:v:20:y:2014:i:2:p:77-100:n:1
    DOI: 10.1515/mcma-2013-0019
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    References listed on IDEAS

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    1. Kurbanmuradov O. & Rannik U. & Sabelfeld K. & Vesala T., 1999. "Direct and Adjoint Monte Carlo Algorithms for the Footprint Problem," Monte Carlo Methods and Applications, De Gruyter, vol. 5(2), pages 85-112, December.
    2. Paolo Guasoni & Scott Robertson, 2008. "Optimal importance sampling with explicit formulas in continuous time," Finance and Stochastics, Springer, vol. 12(1), pages 1-19, January.
    3. Kay Giesecke & Dmitry Smelov, 2013. "Exact Sampling of Jump Diffusions," Operations Research, INFORMS, vol. 61(4), pages 894-907, August.
    4. DiCesare, Joe & Mcleish, Don, 2008. "Simulation of jump diffusions and the pricing of options," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 316-326, December.
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