2013/07/30 by Jim Pitman, Wenpin Tang, Pitman, Jim +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60C05 #60G17 #60J65 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories #math.PR #msc:60C05 #msc:60G17 #msc:60J65
paper · pdf · doi:10.48550/arxiv.1307.7952
26 Pages, 7 figures. Several sections are modified compared to the previous version. The paper is merged into a three-author paper "The Vervaat transform of Brownian bridges and Brownian motion" by Titus Lupu, Jim Pitman and Wenpin Tang
openalex publication_date 2013/07/30 · arxiv created 2013/10/14 · arxiv updated 2013/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 for random walks and Brownian motion, we study the Vervaat transform of Brownian motion and Brownian bridges with arbitary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decompositions and study its semimartingale properties. The expectation and variance of the Vervaat transform of Brownian motion are also derived.