Parisian times for linear diffusions

TL;DR

This paper derives the joint distribution of the first times a linear diffusion exceeds a certain duration above or below a level, extending Parisian barrier option analysis beyond Brownian motion with drift.

Contribution

It introduces new formulas and independence properties for Parisian times in general linear diffusions, broadening the scope beyond classical Brownian models.

Findings

Derived joint distributions for Parisian times in linear diffusions

Established independence properties of these stopping times

Provided formulas for ruin probabilities related to Parisian barriers

Abstract

We compute the joint distribution of the first times a linear diffusion makes an excursion longer than some given duration above (resp. below) some fixed level. In the literature, such stopping times have been introduced and studied in the framework of \emph{Parisian} barrier options, mainly in the case of Brownian motion with drift. We also exhibit several independence properties, and provide some formulae for the associated ruin probabilities.