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