Some distributional properties of a Brownian motion with a drift and an extension of P. Lévy's theorem

The theorem proved by P. Lévy states that $(\sup B-B, \sup B)\stackrel{\mathrm{law}}{=}(|B|,L(B))$. Here, $B$ is a standard linear Brownian motion and $L(B)$ is its local time in zero. In this paper, we present an extension of P. Lévy's theorem to the case of a Brownian motion with a (random) drift as well as to the case of conditionally Gaussian martingales. We also give a simple proof of the equality $2\sup B^{\lambda}-B^{\lambda}\stackrel{\mathrm{law}}{=}|B^{\lambda}|+L(B^{\lambda})$, where $B^{\lambda}$ is the Brownian motion with a drift ${\lambda}\in\mathbb{R}$.