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 $e^{(\mu )}=\{e_t^{(\mu )},t\geqq 0\}$ and its images by certain transforms $T_{\alpha }$ involving $e^{(\mu )}$ and its quadratic variation $\{⟨e^{(\mu )}{⟩}_t,t\geqq 0\}$. 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 $\mu$, 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.