  For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat
transform $V(f)(t):=f(\tau(f)+t \mod1)+f(1)1_{\{t+\tau(f) \geq
1\}}-f(\tau(f))$, where $\tau(f)$ 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.
