The Laws of, and Conditioning with Respect to, Last Passage Times

  • Ju-Yi Yen
  • Marc Yor
Part of the Lecture Notes in Mathematics book series (LNM, volume 2088)


Starting with some simple relationship between the law of a transient Bessel process, Brownian scaled at a last passage time γ, and the law of the Bessel process itself, the law of γ for a general transient diffusion in (0, ) is expressed in terms of the density of its semi-group, considered with respect to time. Conditioning the past of this diffusion with respect to γ = t, is shown to yield the law of the corresponding diffusion bridge of duration t. As an example, the law of the first hit of 0 by an Ornstein–Uhlenbeck process is obtained.


Passage Time Infinitesimal Generator Uhlenbeck Process Bessel Process Area Formula 
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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Ju-Yi Yen
    • 1
  • Marc Yor
    • 2
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA
  2. 2.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris VIParis CX 05France

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