The greatest convex minorant of Brownian motion, meander, and bridge

TL;DR

This paper provides new descriptions and analyses of the greatest convex minorant of Brownian motion, revealing identities and properties of its features through point process and sequential (Markov chain) perspectives.

Contribution

It introduces novel point process and sequential descriptions of the convex minorant, establishing their equivalence and deriving new identities and properties.

Findings

New point process and sequential descriptions of the convex minorant

Identification of identities between derived quantities

Analysis of the Markov chain properties of the sequential description

Abstract

This article contains both a point process and a sequential description of the greatest convex minorant of Brownian motion on a finite interval. We use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the Brownian motion meets its minorant. The equivalence of the these descriptions is non-trivial, which leads to many interesting identities between quantities derived from our analysis. The sequential description can be viewed as a Markov chain for which we derive some fundamental properties.