A Note on Random Walks with Absorbing barriers and Sequential Monte Carlo Methods

TL;DR

This paper provides a detailed variance analysis of importance sampling and sequential Monte Carlo methods applied to one-dimensional random walks with absorbing barriers, offering explicit estimates and stability insights.

Contribution

It introduces a precise variance analysis for IS and SMC methods in this specific model, including explicit spectral formulae and stability properties.

Findings

Derived sharp variance estimates for IS and SMC procedures.

Compared variance properties of SMC and IS techniques.

Provided stability analysis of normalized Feynman-Kac semigroups.

Abstract

In this article we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of 1-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman-Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article, is one of the few to consider an in-depth analysis of an SMC method for a particular model-type as well as variance comparison of SMC algorithms.