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

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 \{\langle e^{(μ)}\rangle_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–Föllmer for solutions of certain stochastic differential equations with anticipative drifts.

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