Random Brownian scaling identities and splicing of Bessel processes. (English) Zbl 0937.60079

Pitman, Jim; Yor, Marc

Ann. Probab. 26, No. 4, 1683-1702 (1998).

Summary: An identity in distribution due to F. B. Knight [Astérisque 157/158, 233-247 (1988; Zbl 0665.60072)] for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of Brownian excursions, due to Williams and Bismut, each involving back-to-back splicing of fragments of two independent three-dimensional Bessel processes. Generalizations of both splicing constructions are described, which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.

MSC:

60J65	Brownian motion
60G18	Self-similar stochastic processes
60J60	Diffusion processes

Keywords:

Brownian bridge; Brownian excursion; Brownian scaling; path transformation; Williams’ decomposition; local time; Bessel process; range process

Citations:

Zbl 0665.60072

References:

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