State-independent Importance Sampling for Random Walks with Regularly Varying Increments

TL;DR

This paper introduces novel state-independent importance sampling algorithms for efficiently estimating rare event probabilities in heavy-tailed random walks, including cases with infinite variance, outperforming existing methods.

Contribution

It proposes a decomposition-based approach to develop simpler, state-independent importance sampling algorithms for heavy-tailed random walks, including infinite variance cases.

Findings

Estimators perform at least as well as existing methods.

Algorithms are effective even with infinite variance increments.

Numerical results show substantial improvements over state-dependent estimators.

Abstract

We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and elegant state-dependent efficient simulation algorithms have been developed in the literature over the last few years. We propose that by suitably decomposing these rare event probabilities into a dominant and further residual components, simpler state-independent importance sampling algorithms can be devised for each component resulting in composite unbiased estimators with desirable efficiency properties. When the increments have infinite variance, there is an added complexity in estimating the level crossing probabilities as even the well known zero-variance measures have an infinite expected termination time. We adapt our algorithms so that this expectation is finite while the estimators remain strongly efficient. Numerically, the proposed estimators perform at least as well, and sometimes substantially better than the existing state-dependent estimators in the literature.