Generalized inverse Gaussian distributions and the time of first level crossing

TL;DR

This paper introduces a new approximation for the first crossing time distribution of a process involving a compound renewal process and a drift, outperforming existing methods especially near critical drift values, and relates it to generalized inverse Gaussian distributions.

Contribution

It presents a novel approximation method for the first crossing time distribution, leveraging generalized inverse Gaussian distributions, with improved accuracy near critical drift points.

Findings

Outperforms existing approximations around the critical point c=𝓢

Provides a tight relation to generalized inverse Gaussian distributions

Enhances understanding of crossing times in renewal processes

Abstract

We propose a new approximation for the distribution of the time of the first crossing of a high level $u$ by random process $\homV{s}-cs$, where $\homV{s}$, $s>0$, is compound renewal process and $c>0$. It significantly outperforms the existing approximations, particularly in the region around the critical point $c=\cS$ which separates processes with positive and negative drifts. This approximation is tightly related to generalized inverse Gaussian distributions.