On the local time process of a skew Brownian motion

Andrei Borodin, Paavo Salminen

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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 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.

    Original languageUndefined/Unknown
    Pages (from-to)3597–3618
    JournalTransactions of the American Mathematical Society
    Volume372
    Issue number5
    DOIs
    Publication statusPublished - 2019
    MoE publication typeA1 Journal article-refereed

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