1. Invariance of Brownian motion associated with past and future maxima
- Author
-
Hariya, Yuu
- Subjects
Mathematics - Probability ,60J65 (Primary) 60J55 (Secondary) - Abstract
Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion. As an application of a recent result of ours on exponential functionals of Brownian motion, we show in this paper that, for every fixed $t>0$, the process given by \begin{align*} B_{s}-B_{t}-\Bigl| B_{t}+\max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u} \Bigr| +\Bigl| \max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u} \Bigr| ,\quad 0\le s\le t, \end{align*} is a Brownian motion. The path transformation that describes the above process is proven to be an involution, commute with time reversal, and preserve Pitman's transformation. A connection with Pitman's $2M-X$ theorem is also discussed., Comment: 20 pages. Proposition 1.4 is added in order to explain a connection with Pitman's 2M-X theorem in Section 4
- Published
- 2023