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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.

Some absolute continuity relationships for certain anticipative transformations of geometric Brownian motions