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### Boundary-crossing identities for diffusions having the time-inversion property

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Alili, Larbi and Patie, P..
(2010)
*Boundary-crossing identities for diffusions having the time-inversion property.*
Journal of Theoretical Probability, Vol.23
(No.1).
pp. 65-84.
ISSN 0894-9840

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WRAP_Alili_alili_patie_#JOTP-241R1.pdf - Draft Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (264Kb) |

Official URL: http://dx.doi.org/10.1007/s10959-009-0245-3

## Abstract

We review and study a one-parameter family of functional transformations, denoted by (S (β)) β∈ℝ, which, in the case β<0, provides a path realization of bridges associated to the family of diffusion processes enjoying the time-inversion property. This family includes Brownian motions, Bessel processes with a positive dimension and their conservative h-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary-crossing times for these diffusions over a given function f to those over the image of f by the mapping S (β), for some fixed β∈ℝ. We give some new examples of boundary-crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family.

Item Type: | Journal Article |
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Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Statistics |

Library of Congress Subject Headings (LCSH): | Brownian motion processes, Self-similar processes, Bessel functions, Diffusion processes, Markov processes |

Journal or Publication Title: | Journal of Theoretical Probability |

Publisher: | Springer New York LLC |

ISSN: | 0894-9840 |

Official Date: | March 2010 |

Volume: | Vol.23 |

Number: | No.1 |

Page Range: | pp. 65-84 |

Identification Number: | 10.1007/s10959-009-0245-3 |

Status: | Peer Reviewed |

Access rights to Published version: | Restricted or Subscription Access |

References: | [1] L. Alili and P. Patie. On the �rst crossing times of a Brownian motion and a family of continuous |

URI: | http://wrap.warwick.ac.uk/id/eprint/3298 |

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