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Zbl 0884.60071
Pitman, Jim; Yor, Marc
On the lengths of excursions of some Markov processes. (English)
[A] Azéma, J. (ed.) et al., Séminaire de probabilités XXXI. Berlin: Springer. Lect. Notes Math. 1655, 272-286 (1997). ISBN 3-540-62634-4/pbk

Summary: Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the Lévy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional diffusion are related to martingales studied by Azéma and Rainer. Formulae are obtained which show how the distribution of interval lengths is affected when the underlying process is subjected to a Girsanov transformation. In particular, results for the zero set of an Ornstein-Uhlenbeck process or a Cox-Ingersoll-Ross process are derived from results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator.

MSC 2000:: *60J27 Markov chains with continuous parameter

Keywords: Lévy measure; Girsanov transformation; Ornstein-Uhlenbeck process; Bessel process

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