Excursions Above the Minimum for Diffusions

TL;DR

This paper establishes a Lévy system for excursions of one-dimensional diffusions above their past minima, providing new proofs for Williams' decomposition and Vervaat's theorem, enhancing understanding of diffusion path structures.

Contribution

It introduces a Lévy system for diffusion excursions and offers direct proofs of key decomposition and theorem, advancing theoretical insights in stochastic processes.

Findings

Lévy system for diffusion excursions established

Direct proofs of Williams' decomposition provided

Vervaat's theorem re-proven with new approach

Abstract

We demonstrate the existence of a "L\'evy system" for the excursions of a one-dimensional diffusion process above its past-minimum process. As applications we provide a direct proof of D. Williams' decomposition (in both a global and a local form) and of a theorem of W. Vervaat.