The local properties of the Markov processes of Ornstein-Uhlenbeck type

TL;DR

This paper investigates the local properties of Ornstein-Uhlenbeck type processes driven by Lévy processes, establishing the existence and continuity of local time, analyzing its asymptotic behavior, and exploring the first passage problem.

Contribution

It proves the existence, continuity, and regularity of local time for Ornstein-Uhlenbeck processes driven by Lévy noise under mild conditions, and examines their asymptotic and first passage properties.

Findings

Existence of local time for Ornstein-Uhlenbeck processes.

Continuity and regularity of local time for almost every x.

Asymptotic behavior of local time in ergodic cases.

Abstract

We prove the existence of a local time, the continuity of the local time about $t$, and the regular property for $a.e.$ $x\in R$ of a Ornstein-Uhlenbeck type $\{X_t,\ t\in R^+\}$ driven by a general L\'{e}vy process, under mild regularity conditions. We discuss the asymptotic behaviour of the local time when $X$ is ergodic. We also investigate the first passage problem. These results give precise information about the local properties of the sample functions.