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

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.

49 pages, 15 figures

References in corpus (16)

• The large deviation approach to statistical mechanics
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Introduction to dynamical large deviations of Markov processes
• Survival of an evasive prey
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Lagrange Inversion
• Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension
• Sparre-Andersen theorem with spatiotemporal correlations
• An introduction to large deviations with applications in physics
• The cost of resetting discrete-time random walks
• First-passage properties of the jump process with a drift. Two exactly solvable cases
• Importance Sampling for counting statistics in one-dimensional systems
• Ladder costs for random walks in Lévy random media
• Numerical Aspects of Large Deviations
• Beyond Poisson: First-Passage Asymptotics of Renewal Shot Noise
• First-passage properties of the jump process with a drift. The general case