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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the local time process of a skew Brownian motion


Authors: Andrei Borodin and Paavo Salminen
Journal: Trans. Amer. Math. Soc. 372 (2019), 3597-3618
MSC (2010): Primary 60J65, 60J60, 60J55
DOI: https://doi.org/10.1090/tran/7852
Published electronically: June 3, 2019
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Abstract: We derive a Ray-Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both of these cases. The Ray-Knight type theorem is applied to study integral functionals of the local time process of the skew Brownian motion. In particular, we determine the distribution of the maximum of the local time process up to a fixed time, which can be seen as the main new result of the paper.


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Additional Information

Andrei Borodin
Affiliation: National Research University, Higher School of Economics, Campus Saint Petersburg, Saint Petersburg, Russia
Email: borodin@pdmi.ras.ru

Paavo Salminen
Affiliation: Faculty of Science and Engineering, Abo Akademi, Turku, Finland
Email: paavo.salminen@abo.fi

DOI: https://doi.org/10.1090/tran/7852
Keywords: Brownian motion, local time, Bessel function, inversion formula for Laplace transforms
Received by editor(s): October 18, 2018
Received by editor(s) in revised form: February 5, 2019
Published electronically: June 3, 2019
Additional Notes: This research was partially supported by a grant from Magnus Ehrnrooths stiftelse, Finland, and grant SPbU-DGF 6.65.37.2017.
Article copyright: © Copyright 2019 American Mathematical Society