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We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) +... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitableMarkov chain. We assume that the Y_j's satisfy the so-called Cramer conditionwith associated root theta_{ast} in (0,infty) and that the tails of the B_j'sare appropriately behaved so that D is regularly varying with indextheta_{ast}. We illustrate by means of an example that the naturalstate-independent importance sampling estimator obtained by exponentiallytilting the Y_j's according to theta_{ast} fails to provide an efficientestimator (in the sense of appropriately controlling the relative mean squarederror as the tail probability of interest gets smaller). Then, we constructestimators based on state-dependent importance sampling that are rigorouslyshown to be efficient.

Efficient Rare-event Simulation for Perpetuities