2013/10/31 by Titus Lupu, Jim Pitman, Wenpin Tang
Mathematics · #math.PR #msc:60C05 #msc:60G17 #msc:60J65
paper · pdf · doi:10.1214/ejp.v20-3744
published as Electro. J. Probab 20 (2015) no.51, 1-31 · 31 Pages, 4 figures. This paper (published by http://ejp.ejpecp.org/article/view/3744) is combined of two papers "On Vervaat transform of Brownian bridges and Brownian motion"(arXiv:1307.7952) by Jim Pitman and Wenpin Tang, and "Some points on Vervaat's transform of Brownian bridges and Brownian motion"(arXiv:1308.3759) by Titus Lupu
arxiv created 2015/05/08 · arxiv updated 2015/05/11
For a continuous function f ∈ C([0,1]), define the Vervaat transform V(f)(t):=f(τ(f)+t \mod1)+f(1)1_\t+τ(f) ≥ 1\-f(τ(f)), where τ(f) corresponds to the first time at which the minimum of f is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.