A first passage problem for a Poisson counting process with a linear moving boundary

TL;DR

This paper unifies and extends the analysis of the first passage time for a Poisson process crossing a linear boundary, providing new exact results and insights into its critical behavior.

Contribution

It offers a unified, pedagogical treatment of existing methods and derives new explicit large deviation functions and closed-form expressions for the first passage time.

Findings

Derived an explicit large deviation function for the first-passage time.

Obtained closed-form expressions for the conditional mean first-passage time.

Revealed non-trivial critical behavior in the crossing process.

Abstract

The time to first crossing for the Poisson counting process with respect to a linear moving barrier with offset is a classic problem, although key results remain scattered across the literature and their equivalence is often unclear. Here we present a unified and pedagogical treatment of two approaches: the direct time-domain approach based on path-decomposition techniques and the Laplace-domain approach based on the Pollaczek-Spitzer formula. Beyond streamlining existing derivations and establishing their consistency, we leverage the complementary nature of the two methods to obtain new exact analytical results. Specifically, we derive an explicit large deviation function for the first-passage time distribution in the subcritical regime and closed-form expressions for the conditional mean first-passage time for arbitrary offset. Despite its simplicity, this first crossing process exhibits non-trivial critical behavior and provides a rare example where all the main results of interest can be derived exactly.