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

In: Itô’s Stochastic Calculus and Probability Theory

Author

Listed:
  • Jim Pitman

    (University of California, Department of Statistics)

  • Marc Yor

    (Université Pierre et Marie Curie, Laboratoire de Probabilités)

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].

Suggested Citation

  • Jim Pitman & Marc Yor, 1996. "Decomposition at the maximum for excursions and bridges of one-dimensional diffusions," Springer Books, in: Nobuyuki Ikeda & Shinzo Watanabe & Masatoshi Fukushima & Hiroshi Kunita (ed.), Itô’s Stochastic Calculus and Probability Theory, pages 293-310, Springer.
    • Handle: RePEc:spr:sprchp:978-4-431-68532-6_19
    DOI: 10.1007/978-4-431-68532-6_19
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