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

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.

A dichotomy for sampling barrier-crossing events of random walks with regularly varying tails

“…The threshold 3/2 also arises in the simulation literature involving barrier-crossing events with regularly varying step sizes [9], [19]. The nature of the threshold we obtain here is, however, different from these works for three reasons.…”

“…Review. Previous works in which the 3/2 threshold appears in the context of efficiency are Blanchet and Liu [9] and Murthy et al [19]. In both papers, the authors focus on solving the probability estimation problem via importance sampling; that is, their aim is to estimate the probability P(τ b < ∞) for arbitrarily large barriers b, using Monte Carlo sampling from another measure.…”

“…Blanchet and Liu proposed a parameterized and state-dependent change of measure, say Q BL , which in the regularly varying case takes the form of a mixture between a big-and a small-jump transition kernel. Murthy et al [19] proposed a similar big-and smalljump mixture kernel, say Q MJB , however their change of measure is state-independent and additionally conditions on the time interval at which the barrier-crossing event occurs.…”

“…For the probability estimating problem under heavy tails, early approaches can be found in [2], [5], and [17]. An important contribution for the current paper is [8], which was later followed by [9] and [19]. A recent new technique can be found in [16], in which Markov Chain Monte Carlo was used to estimate the multiplicative inverse of the probability of crossing the barrier.…”

“…The problem of exact sampling of paths with heavy tails, on the other hand, has only recently been tackled; see [11]. In this paper the authors modified the measure of [19], which focused on the probability estimation problem, and built on the scheme for exact sampling of paths introduced in [10, Section 4]. The approach studied in this paper is based on the Blanchet-Glynn change of measure, which is conceptually simpler than the approach proposed in [11].…”