Poisson-Driven Stationary Markov Models

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-type transform modulating the dependence structure in the model. An appealing feature of our approach is the possibility to control the underlying transition probabilities and, therefore, incorporate them within standard estimation methods. Given the resulting representation of the transition density, a Gibbs sampler algorithm based on the slice method is proposed and implemented. In the discrete-time case, special attention is placed to the class of generalized inverse Gaussian distributions. In the continuous case, we first provide a brief treatment of the class of gamma distributions, and then extend it to cover other invariant distributions, such as the generalized extreme value class. The proposed approach and estimation algorithm are illustrated with real financial datasets. Supplementary materials for this article are available online.

On a flexible construction of a negative binomial model

“…While some appealing families of diffusion processes have been found to have representation (1), under the above extension to continuous time (see, e.g. Anzarut et al, 2018), in general it is not always easy to find analytical conditions that meet Chapman-Kolmogorov equations. Here, we unveil the conditions to apply this construction to negative-binomial marginal distributions, which, as we will see, represent a rich class of continuous-time Markov chains.…”