Some Absolute Continuity Relationships for Certain Anticipative Transformations of Geometric Brownian Motions
Catherine Donati-Martin
Université Paris VI et VII, FranceMarc Yor
Université Paris VI, FranceHiroyuki Matsumoto
Nagoya University, Japan
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Abstract
We present some absolute continuity relationships between the probability laws of a geometric Brownian motion and its images by certain transforms involving and its quadratic variation . 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 theorem for , a Brownian motion with constant drift , and its one-sided supremum . 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.
Cite this article
Catherine Donati-Martin, Marc Yor, Hiroyuki Matsumoto, Some Absolute Continuity Relationships for Certain Anticipative Transformations of Geometric Brownian Motions. Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, pp. 295–326
DOI 10.2977/PRIMS/1145477226