First-passage probabilities and invariant distributions of Kac-Ornstein-Uhlenbeck processes

Abstract

In this paper, we study Ornstein-Uhlenbeck processes with Markov modulation, whose parameters depend on an external underlying two-state Markov process ε. Conditional mean and variance of such processes under given modulation are investigated from the point of view of the first passage probabilities and invariant measures. It is also studied the limiting behaviour under scaling conditions similar to Kac's scaling.