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On functionals of excursions for Bessel processes with negative index

Volume 246 / 2019

Tomasz Byczkowski, Jacek Jakubowski, Maciej Wiśniewolski Studia Mathematica 246 (2019), 217-231 MSC: 60J25, 60J35, 60J45. DOI: 10.4064/sm170309-15-3 Published online: 8 October 2018

Abstract

A closed formula is given for the integral of functionals on the space of excursions from a point $z\geq 0$ of a Bessel process $R^{(\mu )}$ with index $\mu \in (-1,0)$ with respect to the characteristic (Itô) measure. This integral is an integral with respect to some kernel $K$. The kernel $K$ is found due to the explicit form of the joint distribution of $(\tau _z, R_t^{(\mu )})$ at fixed time $t$ and for $\tau _z$ being the first hitting time of the point $z$ by the process $R^{(\mu )}$. The expected time spent by an excursion in an interval with respect to the Itô measure is calculated.

Authors

  • Tomasz ByczkowskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
    e-mail
  • Jacek JakubowskiInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail
  • Maciej WiśniewolskiInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland
    e-mail

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