An extension of P. Lévy's distributional

Abstract

We extend the well-known P. Lévy theorem on the distributional identity $(M_t-B_t,M_t)simeq\left(\right|B_t|,L(B{)}_t)$ , where $\left(B_t\right)$ is a standard Brownian motion and $\left(M_t\right)=\left({\sup }_{0\le s\le t}{B}_s\right)$ to the case of Brownian motion with drift λ. Processes of the type

$$d{X}_t^{\lambda }=-\lambda sgn\left(X_t^{\lambda }\right)dt+d{B}_t$$

appear naturally in the generalization.