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Decomposition at the maximum for excursions and bridges of one-dimensional diffusions

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Itô’s Stochastic Calculus and Probability Theory

Abstract

In his fundamental paper [25], Itô showed how to construct a Poisson point process of excursions of a strong Markov process X over time intervals when X is away from a recurrent point a of its statespace. The point process is parameterized by the local time process of X at a. Each point of the excursion process is a path in a suitable space of possible excursions of X,starting at a at time 0, and returning to a for the first time at some strictly positive time ζ, called the lifetime of the excursion. The intensity measure of the Poisson process of excursions is a σ-finite measure Λ on the space of excursions, known as Itô’s excursion law. Accounts of Itô’s theory of excursions can now be found in several textbooks [48, 46, 10]. His theory has also been generalized to excursions of Markov processes away from a set of states [34, 19, 10] and to excursions of stationary, not necessarily Markovian processes [38].

Research supported in part by N.S.F. Grants MCS91-07531 and DMS-9404345

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Author information

Authors and Affiliations

  1. Department of Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA, 94720-3860, USA

    Jim Pitman

  2. Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu - Tour 56, 75252, Paris Cedex 05, France

    Marc Yor

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  1. Jim Pitman
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Editors and Affiliations

  1. Department of Computer Science, Ritsumeikan University, 525-77, Kusatsu, Shiga, Japan

    Nobuyuki Ikeda (Professor) (Professor)

  2. Department of Mathematics, Graduate School of Science, Kyoto University, 606-01, Kyoto, Japan

    Shinzo Watanabe (Professor) (Professor)

  3. Department of Mathematics, Faculty of Fundamental Engineering, Osaka University, Toyonaka, Osaka, 560, Japan

    Masatoshi Fukushima (Professor) (Professor)

  4. Graduate School of Mathematics, Kyushu University, Fukuoka, 812, Japan

    Hiroshi Kunita (Professor) (Professor)

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© 1996 Springer-Verlag Tokyo

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Pitman, J., Yor, M. (1996). Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In: Ikeda, N., Watanabe, S., Fukushima, M., Kunita, H. (eds) Itô’s Stochastic Calculus and Probability Theory. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68532-6_19

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  • DOI: https://doi.org/10.1007/978-4-431-68532-6_19

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-68534-0

  • Online ISBN: 978-4-431-68532-6

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