First-Exit Times of an Inverse Gaussian Process

TL;DR

This paper analyzes the first-exit time of an inverse Gaussian process, deriving its distribution, exploring its properties, and connecting it to fractional PDEs and subordinated processes, with implications for stable subordinators.

Contribution

It provides explicit distribution functions for the first-exit time of an inverse Gaussian process and links these to fractional PDEs and subordination frameworks, extending previous results.

Findings

Distribution functions are not infinitely divisible.

Tail probabilities decay exponentially.

Distribution functions relate to supremum of Brownian motion with drift.

Abstract

The first-exit time process of an inverse Gaussian L\'evy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail probabilities decay exponentially. These distribution functions can also be viewed as distribution functions of supremum of the Brownian motion with drift. The density function is shown to solve a fractional PDE and the result is also generalized to tempered stable subordinators. The subordination of this process to the Brownian motion is considered and the underlying PDE of the subordinated process is obtained. The infinite divisibility of the first-exit time of a $\beta$-stable subordinator is also discussed.