Markovian structure in the concave majorant of Brownian motion

TL;DR

This paper uncovers a hidden Markovian structure in the concave majorant of Brownian motion, revealing distributional identities and connections to Bessel processes, which enhance understanding of Brownian path properties.

Contribution

It identifies a Markovian structure in the concave majorant of Brownian motion and establishes distributional identities linking it to Bessel processes, providing new insights into stochastic process behavior.

Findings

Distribution of 2K_t - B_t matches that of R_5(t)

Shared properties between 2K - B and R_5

Distributional description of convex minorant of 3D Bessel process with drift

Abstract

The purpose of this paper is to highlight some hidden Markovian structure of the concave majorant of the Brownian motion. Several distributional identities are implied by the joint law of a standard one-dimensional Brownian motion $B$ and its almost surely unique concave majorant $K$ on $[0,\infty)$. In particular, the one-dimensional distribution of $2 K_t - B_t$ is that of $R_5(t)$, where $R_5$ is a $5-$dimensional Bessel process with $R_5(0) = 0$. The process $2K-B$ shares a number of other properties with $R_5$, and we conjecture that it may have the distribution of $R_5$. We also describe the distribution of the convex minorant of a three-dimensional Bessel process with drift.