Convex minorants of random walks and L\'evy processes

TL;DR

This paper reviews recent advances in understanding the convex minorant of random walks and Lévy processes, highlighting point process descriptions and invariance properties that extend existing literature.

Contribution

It provides a comprehensive overview of the latest results on convex minorants, including descriptions based on path transformations and special properties of Brownian motion.

Findings

Point process descriptions for convex minorants on finite intervals

Invariance under path transformations explains these descriptions

Special properties of Brownian motion lead to sequential descriptions

Abstract

This article provides an overview of recent work on descriptions and properties of the convex minorant of random walks and L\'evy processes which summarize and extend the literature on these subjects. The results surveyed include point process descriptions of the convex minorant of random walks and L\'evy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander.