Some Distributional Properties of a Brownian Motion with a Drift and an Extension of P. Lévy's Theorem

Abstract

The theorem proved by P. Levy states that $(\sup B-B,\,\sup B){\;\stackrel{\rm law}{=}\;}(|B|,\,L(B))$. 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. Levy'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 $2\sup B^{\lambda}-B^{\lambda}{\;\stackrel{\rm law}{=}\;}|B^{\lambda}|+L(B^{\lambda})$, where $B^{\lambda}$ is the Brownian motion with a drift ${\lambda}\in\bR$.