Efficient rare-event simulation for perpetuities

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 suitable Markov chain. We assume that the Y j ’s satisfy the so-called Cramér condition with associated root θ ∗ ∈ ( 0 , ∞ ) and that the tails of the B j ’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 Y j ’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.