JournalsprimsVol. 37, No. 3pp. 295–326

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