On fluctuation-theoretic decompositions via Lindley-type recursions

TL;DR

This paper presents a novel fluctuation-theoretic decomposition for the joint distribution of the maximum of a Lévy process and its last occurrence time, using Lindley-type recursions and Poisson observations.

Contribution

It introduces a new distributional equality decomposing the maximum and last occurrence time into independent components, extending Lindley recursion insights.

Findings

The decomposition holds for Lévy processes over exponential times.

Elementary proof provides deeper understanding of the decomposition.

Generalizations of Lindley recursion are derived from the proof technique.

Abstract

Consider a L\'evy process $Y(t)$ over an exponentially distributed time $T_\beta$ with mean $1/\beta$. We study the joint distribution of the running maximum $\bar{Y}(T_\beta)$ and the time epoch $G(T_\beta$) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector ($\bar{Y}(T_\beta),G(T_\beta)$) can be written as a sum of two independent vectors, the first one being ($\bar{Y}(T_{\beta+\omega}),G(T_{\beta+\omega})$) and the second one being the running maximum and corresponding time epoch under the restriction that the L\'evy process is only observed at Poisson($\omega$) inspection epochs (until $T_\beta$). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that $\omega$ only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.