On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein-Uhlenbeck processes

TL;DR

This paper derives explicit formulas for moments and stationary distribution characteristics of Markov modulated Ornstein-Uhlenbeck processes, based on new results for Markov additive processes and their integrals.

Contribution

It provides novel explicit formulas for moments and stationary distributions of Markov modulated Ornstein-Uhlenbeck processes, extending understanding of their probabilistic properties.

Findings

Explicit formulas for the $$'th moments of the processes.

Characterization of stationary distribution moments and autocovariance.

New general results on moments of Markov additive processes.

Abstract

We establish sufficient conditions for the existence, and derive explicit formulas for the $\kappa$'th moments, $\kappa \geq 1$, of Markov modulated generalized Ornstein-Uhlenbeck processes as well as their stationary distributions. In particular, the running mean, the autocovariance function, and integer moments of the stationary distribution are derived in terms of the characteristics of the driving Markov additive process. Our derivations rely on new general results on moments of Markov additive processes and (multidimensional) integrals with respect to Markov additive processes.