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.

Optimal rare event Monte Carlo for Markov modulated regularly varying random walks

“…For the case of i.i.d. increments, we have a parametrized state-dependent sampler for the estimation of P{S n > b} due to (Blanchet and Liu 2008), and two recent state-independent algorithms in (Murthy and Juneja 2012) and (Murthy, Juneja, and Blanchet 2013). In (Blanchet and Liu 2008), the parameters to sample increments based on the current position of the walk is carefully done in every step by ensuring that a certain Lyapunov inequality holds; here the fact that each increment has zero mean has been used crucially to enforce the Lyapunov inequality.…”

“…One more interesting aspect of our importance sampling changes of measure is that it does not involve any change in the transition probabilities of the modulating Markov chain, unlike the case of modulated walks with light-tailed increments where exponential twisting is performed to favour certain states over others. As in (Murthy, Juneja, and Blanchet 2013), we partition the event of interest into a dominant and residual component, and prescribe simple, intuitive sampling measures, that are easy to implement and offer a wide scope for generalizing to complex multi-dimensional settings.…”