On the excursion theory for linear diffusions

TL;DR

This paper develops new identities in the excursion theory of linear diffusions, analyzing excursions around exponential times and connecting stationary and non-stationary cases, with applications to Ornstein-Uhlenbeck processes.

Contribution

It introduces novel identities for excursions of linear diffusions, especially for excursions straddling exponential times, and characterizes their laws using Krein's representations.

Findings

The law of the excursion length is infinitely divisible.

Connections established between stationary and non-stationary diffusion excursions.

Application to Ornstein-Uhlenbeck processes demonstrates practical relevance.

Abstract

We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein's representations that, e.g., the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss Ornstein-Uhlenbeck processes.