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Efficient rare-event simulation for perpetuities

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  • Blanchet, Jose
  • Lam, Henry
  • Zwart, Bert
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    Abstract

    We consider perpetuities of the form D=B1exp(Y1)+B2exp(Y1+Y2)+⋯, where the Yj’s and Bj’s might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the Yj’s satisfy the so-called Cramér condition with associated root θ∗∈(0,∞) and that the tails of the Bj’s are appropriately behaved so that D is regularly varying with index θ∗. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the Yj’s according to θ∗ fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 10 ()
    Pages: 3361-3392

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:10:p:3361-3392

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    Related research

    Keywords: State-dependent importance sampling; Perpetuities; Tail asymptotics; Lyapunov inequalities; Markov chains;

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    1. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
    2. de Saporta, BenoI^te, 2005. "Tail of the stationary solution of the stochastic equation Yn+1=anYn+bn with Markovian coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1954-1978, December.
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