Ornstein-Uhlenbeck processes of bounded variation

TL;DR

This paper introduces a bounded variation Ornstein-Uhlenbeck process driven by a telegraph process, providing explicit distributions, moments, and demonstrating convergence to the classical process under Kac's rescaling.

Contribution

It presents a novel Ornstein-Uhlenbeck process with bounded variation driven by a telegraph process, including explicit distributional results and limit behavior analysis.

Findings

Explicit distribution of the process's hitting time is derived.

Mean and variance formulas are obtained based on joint distributions.

Under Kac's rescaling, the process converges to the classical Ornstein-Uhlenbeck process.

Abstract

Ornstein-Uhlenbeck process of bounded variation is introduced as a solution of an analogue of the Langevin equation with an integrated telegraph process replacing a Brownian motion. There is an interval $I$ such that the process starting from the internal point of $I$ always remains within $I$. Starting outside, this process a. s. reaches this interval in a finite time. The distribution of the time for which the process falls into this interval is obtained explicitly. The certain formulae for the mean and the variance of this process are obtained on the basis of the joint distribution of the telegraph process and its integrated copy. Under Kac's rescaling, the limit process is identified as the classical Ornstein-Uhlenbeck process.