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Article
Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude
Studia Sci. Math. Hungar. (1999)
  • Jim Pitman, University of California, Berkeley
  • Marc Yor
Abstract

We give two new proofs of Csaki's formula for the law of the ratio 1-Q of the maximum relative to the amplitude (i.e. the maximum minus minimum) for a standard Brownian bridge. The second of these proofs is based on an absolute continuity relation between the law of the Brownian bridge restricted to the event (Q < v) and the law of a process obtained by a Brownian scaling operation after back-to back joining of two independent three-dimensional Bessel processes, each started at v and run until it first hits 1. Variants of this construction and some properties of the joint law of Q and the amplitude are described.

Keywords
  • path decomposition,
  • Brownian bridge,
  • maximum,
  • amplitude
Publication Date
1999
Citation Information
Jim Pitman and Marc Yor. "Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude" Studia Sci. Math. Hungar. Vol. 35 Iss. 520 (1999)
Available at: http://works.bepress.com/jim_pitman/6/