Some Absolute Continuity Relationships for Certain Anticipative Transformations of Geometric Brownian Motions Klein-Gordon equations is studied in the Sobolev space Hs = Hs(

Abstract

We present some absolute continuity relationships between the probability laws of a geometric Brownian motion e(μ) = {e__t(μ), t ≧ 0} and its images by certain transforms T_α involving e(μ) and its quadratic variation {<e(μ)>t, t ≧ 0}. These results are derived from, and shown to be closely related to, our previous results about the generalized Dufresne’s identity and the exponential type extensions of Pitman’s 2_M_−_X theorem for X, a Brownian motion with constant drift μ, and its one-sided supremum M. These absolute continuity results are then shown to be particular cases of those by Ramer–Kusuoka for non-linear transformations of the Wiener space and by Buckdahn–Föllmer for solutions of certain stochastic differential equations with anticipative drifts.