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Calculating Principal Eigen-Functions of Non-Negative Integral Kernels: Particle Approximations and Applications

Author

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  • Nick Whiteley

    (School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom)

  • Nikolas Kantas

    (Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom)

Abstract

Often in applications such as rare events estimation or optimal control it is required that one calculates the principal eigenfunction and eigenvalue of a nonnegative integral kernel. Except in the finite-dimensional case, usually neither the principal eigenfunction nor the eigenvalue can be computed exactly. In this paper, we develop numerical approximations for these quantities. We show how a generic interacting particle algorithm can be used to deliver numerical approximations of the eigenquantities and the associated so-called “twisted” Markov kernel as well as how these approximations are relevant to the aforementioned applications. In addition, we study a collection of random integral operators underlying the algorithm, address some of their mean and pathwise properties, and obtain error estimates. Finally, numerical examples are provided in the context of importance sampling for computing tail probabilities of Markov chains and computing value functions for a class of stochastic optimal control problems.

Suggested Citation

  • Nick Whiteley & Nikolas Kantas, 2017. "Calculating Principal Eigen-Functions of Non-Negative Integral Kernels: Particle Approximations and Applications," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1007-1034, November.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:4:p:1007-1034
    DOI: 10.1287/moor.2016.0834
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    References listed on IDEAS

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    1. 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: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    2. repec:dau:papers:123456789/11523 is not listed on IDEAS
    3. Douc, R. & Fort, G. & Moulines, E. & Priouret, P., 2009. "Forgetting the initial distribution for Hidden Markov Models," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1235-1256, April.
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    Cited by:

    1. Angeli, Letizia & Grosskinsky, Stefan & Johansen, Adam M., 2021. "Limit theorems for cloning algorithms," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 117-152.

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