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Efficient rare-event simulation for perpetuities
Authors:Jose Blanchet  Henry Lam  Bert Zwart
Institution:1. Department of Industrial Engineering and Operations Research, Columbia University, United States;2. Department of Mathematics and Statistics, Boston University, United States;3. Probability and Stochastic Networks Group, Centrum Wiskunde & Informatica (CWI), Netherlands
Abstract:We consider perpetuities of the form
D=B1exp(Y1)+B2exp(Y1+Y2)+?,D=B1exp(Y1)+B2exp(Y1+Y2)+?,
where the YjYj’s and BjBj’s might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the YjYj’s satisfy the so-called Cramér condition with associated root θ∈(0,∞)θ(0,) and that the tails of the BjBj’s are appropriately behaved so that DD is regularly varying with index θθ. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the YjYj’s according to θθ 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.
Keywords:State-dependent importance sampling  Perpetuities  Tail asymptotics  Lyapunov inequalities  Markov chains
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