DISTANCE BETWEEN TWO SKEW BROWNIAN MOTIONS AS A SDE WITH JUMPS AND LAW OF THE HITTING TIME

• 成果类型: Article
• 署名作者: Gloter, Arnaud; Martinez, Miguel
• 署名单位: Universite Paris Saclay; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP776
• 发表日期: 2013
• 页码: 1628-1655
• 关键词: flow

摘要

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.