  In a recent work J. Pitman and W. Tang defined the Vervaat's transform for a
Brownian bridge with two different endpoints and for a Brownian motion between
times $0$ and $1$. They proved some path decomposition properties for these
Vervaat's transforms and raised open questions on their semi-martingale
decomposition. In this paper we give an alternative proof for the path
decomposition and answer the open questions.
