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

Some points on Vervaat's transform of Brownian bridges and Brownian motion