On the time of first level crossing and inverse Gaussian distribution

TL;DR

This paper introduces a new approximation for the distribution of the first level crossing time of a compound renewal process with drift, closely related to inverse Gaussian distributions, especially near a critical drift point.

Contribution

It presents a novel approximation method for the crossing time distribution that outperforms existing methods around the critical drift point, linking it to inverse Gaussian distributions.

Findings

The approximation is competitive with existing methods.

It is particularly accurate near the critical drift point c=𝓢.

The method is related to inverse Gaussian distributions.

Abstract

We propose a new approximation for the distribution of the time of the first level $u$ crossing by the random process $\homV{s}-cs$, where $\homV{s}$, $s>0$, is compound renewal process and $c>0$. It is competitive with respect to 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 inverse Gaussian distributions.