# Some Absolute Continuity Relationships for Certain Anticipative Transformations of Geometric Brownian Motions Klein-Gordon equations is studied in the Sobolev space Hs = Hs(

### Catherine Donati-Martin

Université Paris VI et VII, France### Marc Yor

Université Paris VI, France### Hiroyuki Matsumoto

Nagoya University, Japan

## 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 {<e(μ)>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 2_M_−_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.

## Cite this article

Catherine Donati-Martin, Marc Yor, Hiroyuki Matsumoto, Some Absolute Continuity Relationships for Certain Anticipative Transformations of Geometric Brownian Motions Klein-Gordon equations is studied in the Sobolev space Hs = Hs(. Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, pp. 295–326

DOI 10.2977/PRIMS/1145477226