Markovian structure in the concave majorant of Brownian motion
Mehdi Ouaki, Jim Pitman · Electronic Journal of Probability · 2022
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,∞). In particular, the one-dimensional distribution of 2Kt−Bt is that of R5(t), where R5 is a 5-dimensional Bessel process with R5(0)=0. The process 2K−B shares a number of other properties with R5, and we conjecture that it may have the distribution of R5. We also describe the distribution of the convex minorant of a three-dimensional Bessel process with drift.