Denote by αt (μ) the probability law of A t (μ) =∫0 t exp(2(B s +μ s))ds for a Brownian motion{B s , s ≥ 0}. It is well known that αt (μ) is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt (0) andαt (1), as well as an equally remarkable relationship betweenαt (μ) andαt (ν) for two different drifts μ and ν. In this paper, hinging on previous results about αt (μ), we give different proofs of Dufresne's results and present extensions of them for the processes{A t (μ), t ≥ 0}.
