Markovian structure in the concave majorant of Brownian motion

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.