Efficient Rare-event Simulation for Perpetuities

TL;DR

This paper develops efficient state-dependent importance sampling methods for estimating rare tail probabilities of perpetuities, overcoming limitations of traditional state-independent approaches.

Contribution

It introduces novel state-dependent importance sampling estimators that are proven to be efficient for perpetuities with heavy tails, improving upon existing methods.

Findings

State-independent importance sampling fails for certain perpetuities.

State-dependent estimators are rigorously shown to be efficient.

The methods effectively estimate rare event probabilities in heavy-tailed models.

Abstract

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 Cramer condition with associated root theta_{ast} in (0,infty) and that the tails of the B_j's are appropriately behaved so that D is regularly varying with index theta_{ast}. 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 theta_{ast} 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.