Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time
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Arnaud Gloter [1] ; Miguel Martinez [2]
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[1] University of Évry Val d'Essonne
University of Évry Val d'Essonne
Arrondissement d'Évry, Francia
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[2] University of Marne la Vallée
University of Marne la Vallée
París, Francia
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 3, 1, 2013, págs. 1628-1655
- Idioma: inglés
- DOI: 10.1214/12-AOP776
- Texto completo no disponible (Saber más ...)
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Resumen
In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two processes with a stochastic differential equation. This S.D.E. possesses a jump component driven by the excursion process of one of the two skew Brownian motions. Using this representation, we show that the local time of two skew Brownian motions at their first hitting time is distributed as a simple function of a Beta random variable. This extends a result by Burdzy and Chen [Ann. Probab. 29 (2001) 1693–1715], where the law of coalescence of two skew Brownian motions with the same skewness coefficient is computed.
