On Decomposition of the Last Passage Time of Diffusions

TL;DR

This paper derives a new explicit decomposition formula for the last passage time of a regular transient diffusion to a specific state, using transformations involving occupation times and Green functions.

Contribution

It introduces a novel decomposition of the last passage time for diffusions, expressing it through Green functions and occupation time transformations.

Findings

Derived explicit Laplace transform decomposition formula

Established Green function decomposition formula

Applied formulas to diffusion with two-valued parameters

Abstract

For a regular transient diffusion, we provide a decomposition of its last passage time to a certain state $\alpha$. This is accomplished by transforming the original diffusion into two diffusions using the occupation time of the area above and below $\alpha$. Based on these two processes, both having a reflecting boundary at $\alpha$, we derive the decomposition formula of the Laplace transform of the last passage time explicitly in a simple form in terms of Green functions. This equation also leads to the Green function's decomposition formula. We demonstrate an application of these formulas to a diffusion with two-valued parameters.