  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.
