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

TL;DR

This paper analyzes the first-passage probabilities and invariant distributions of Markov-modulated Ornstein-Uhlenbeck processes, exploring their limiting behavior under specific scaling conditions, providing new insights into their stochastic properties.

Contribution

It introduces a detailed study of first-passage probabilities and invariant measures for Markov-modulated Ornstein-Uhlenbeck processes, including their asymptotic behavior under Kac-like scaling.

Findings

Derived explicit formulas for first-passage probabilities.

Characterized invariant distributions of the processes.

Analyzed the limiting behavior under scaling conditions.

Abstract

In this paper, we study Ornstein-Uhlenbeck processes with Markov modulation, whose parameters depend on an external underlying two-state Markov process. 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.