The vervaat transform of brownian bridges and brownian motion

Abstract: For a continuous function f ∈ C([0, 1]), define the Vervaat transform V (f)(t): = f(τ(f) + t mod 1) + 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 study the Vervaat trans-form 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 de-scribe its distribution by path decomposition and study its semimartingale property. Th