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Efficient rare-event simulation for perpetuities

Author

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

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.

Suggested Citation

  • Blanchet, Jose & Lam, Henry & Zwart, Bert, 2012. "Efficient rare-event simulation for perpetuities," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3361-3392.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:10:p:3361-3392
    DOI: 10.1016/j.spa.2012.05.002
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    References listed on IDEAS

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    1. 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.
    2. Paul Dupuis & Hui Wang, 2007. "Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 723-757, August.
    3. 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.
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    1. de Vries, Harwin & Duijzer, Evelot, 2017. "Incorporating driving range variability in network design for refueling facilities," Omega, Elsevier, vol. 69(C), pages 102-114.
    2. Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.

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