In this article we consider Bayesian parameter inference associated to
partially-observed stochastic processes that start from a set B0 and are
stopped or killed at the first hitting time of a known set A. Such processes
occur naturally within the context of a wide variety of applications. The
associated posterior distributions are highly complex and posterior parameter
inference requires the use of advanced Markov chain Monte Carlo (MCMC)
techniques. Our approach uses a recently introduced simulation methodology,
particle Markov chain Monte Carlo (PMCMC) (Andrieu et. al.