On Some Stationary Models: Construction and Estimation

Abstract

We propose a simple yet powerful method to construct strictly stationary Markovian models with given, but arbitrary, invariant distributions. The idea is based on a Poisson transform modulating the dependence structure in the model. An appealing feature of our approach is that we are able to fully control the underlying transition probabilities and therefore incorporate them within standard estimation methods. We analyze some specific cases in both discrete and continuous time. Given our proposed representation of the transition density, a Gibbs sampler algorithm, based on the slice method, is proposed and implemented. In particular, the resulting methodology is of interest for the estimation of certain continuous time models, such as diffusion processes.