An Extension of P. Lévy's Distributional Properties to the Case of a Brownian Motion with Drift

Abstract

We extend the well-known P. Levy theorem on the distributional identity ( M t-B t,M t)simeq(|B t|,L(B) t) , where ( B t) is a standard Brownian motion and ( M t)=(sup 0 ≤s≤tB s) to the case of Brownian motion with drift λ. Processes of the type d X t λ=-λsgn(X t λ)dt+dB t appear naturally in the generalization.