The Annals of Probability

Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time

Arnaud Gloter and Miguel Martinez

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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.

Article information

Ann. Probab., Volume 41, Number 3A (2013), 1628-1655.

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J55: Local time and additive functionals 60J65: Brownian motion [See also 58J65]

Skew Brownian motion local time excursion process Dynkin’s formula


Gloter, Arnaud; Martinez, Miguel. Distance between two skew Brownian motions as a S.D.E. with jumps and law of the hitting time. Ann. Probab. 41 (2013), no. 3A, 1628--1655. doi:10.1214/12-AOP776.

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