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Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion Theory

  • Paavo Salminen
  • Ju-Yi Yen
  • Marc Yor
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

In this note we present integral representations of the Itô excursion measure associated with a general one-dimensional diffusion X. These representations and identities are natural extensions of the classical ones for reflected Brownian motion, RBM. As is well known, the three-dimensional Bessel process, BES(3), plays a crucial rôle in the analysis of the Brownian excursions. Our main interest is in showing explicitly how certain excursion theoretical formulae associated with the pair (RBM, BES(3)) generalize to pair (X, X ), where X denotes the diffusion obtained from X by conditioning X not to hit 0. We illustrate the results for the pair \((R_{-},R_{+})\) consisting of a recurrent Bessel process with dimension \(d_{-} = 2(1-\alpha ),\) α ∈ (0, 1), and a transient Bessel process with dimension \(d_{+} = 2(1+\alpha )\). Pair (RBM, BES(3)) is, clearly, obtained by choosing \(\alpha = 1/2.\)

Notes

Acknowledgements

Paavo Salminen’s research was funded in part by a grant from Svenska kulturfonden via Stiftelsernas professorspool, Finland.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Science and EngineeringÅbo Akademi UniversityÅboFinland
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

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