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Some Absolute Continuity Relationships for Certain Anticipative Transformations of Geometric Brownian Motions

We present some absolute continuity relationships between the probability laws of a geometric Brownian motion e = {e t , t 0} and its images by certain transforms Tα involving e (μ) and its quadratic variation {〈e〉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 2M −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–Follmer for solutions of certain stochastic differential equations with anticipative drifts.