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Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae

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  • Whiteley, Nick
  • Kantas, Nikolas
  • Jasra, Ajay

Abstract

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance for particle approximations of time-homogeneous Feynman–Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman–Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: (1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and (2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.

Suggested Citation

  • Whiteley, Nick & Kantas, Nikolas & Jasra, Ajay, 2012. "Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1840-1865.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1840-1865
    DOI: 10.1016/j.spa.2012.02.002
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    References listed on IDEAS

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    1. Chopin, N. & Del Moral, P. & Rubenthaler, S., 2011. "Stability of Feynman-Kac formulae with path-dependent potentials," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 38-60, January.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters,in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    3. van Handel, Ramon, 2009. "Uniform time average consistency of Monte Carlo particle filters," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3835-3861, November.
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    Cited by:

    1. Persing, Adam & Jasra, Ajay, 2013. "Likelihood computation for hidden Markov models via generalized two-filter smoothing," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1433-1442.

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