Non-Markovianity of $2K-B$ and a degeneration

TL;DR

This paper investigates the process 2K-B, disproves a conjecture about its Markovian nature, and explores its relation to BES processes through degeneration analysis and path properties.

Contribution

It demonstrates that 2K-B is not Markovian, provides a degeneration analysis showing it as a mixture of BES(3), and studies its properties and filtrations.

Findings

2K-B is not a Markov process.

Degeneration of 2K-B is a mixture of BES(3).

Analyzed path decomposition and infinitesimal generator.

Abstract

We study the process of $2K-B$, where $B$ is a standard one-dimensional Brownian motion and $K$ is its concave majorant. In light of Pitman's $2M-B$ theorem, it was recently conjectured by Ouaki and Pitman \cite{OP} that $2K-B$ has the law of the BES(5) process. The two processes share properties such as Brownian scaling, time inversion and quadratic variation, and the same one point distribution and infinitesimal generator, among many other evidences; and it remains to prove that $2K-B$ is Markovian. However, we show that this conjecture is false. To better understand the similarity between these two processes, we study a degeneration of $2K-B$. We show it is a mixture of BES(3), and get other properties including multiple points distribution, infinitesimal generator, and path decomposition at future infimum. We also further investigate the Markovian structure and the filtrations of $2K-B, B$ and $K$.