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Theory of Probability and Mathematical Statistics

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Properties of integrals with respect to fractional Poisson processes with compact kernels


Authors: Y. Mishura and V. Zubchenko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 143-152
MSC (2010): Primary 60G22; Secondary 60G51
DOI: https://doi.org/10.1090/S0094-9000-2015-00941-8
Published electronically: January 26, 2015
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Abstract: Properties of a fractional Poisson process with the Molchan-Golosov kernel are studied. The kernel can be viewed as compact since it is non-zero on a compact interval. The integral of a nonrandom function with respect to centered and non-centered fractional Poisson processes with the Molchan-Golosov kernel is introduced. The second moments of these integrals are obtained in terms of the norm of the integrand in the space $ L_{1/H}([0,T])$. Moment estimates for higher moments of these integrals are established by using the Bichteler-Jacod inequality.


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Additional Information

Y. Mishura
Affiliation: Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty for Mechanics and mathematics, Kyiv National Taras Shevchenko University, Volodymyrska St., 64, Kyiv, 01601, Ukraine
Email: myus@univ.kiev.ua

V. Zubchenko
Affiliation: Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty for Mechanics and mathematics, Kyiv National Taras Shevchenko University, Volodymyrska St., 64, Kyiv, 01601, Ukraine
Email: v_zubchenko@ukr.net

DOI: https://doi.org/10.1090/S0094-9000-2015-00941-8
Keywords: Fractional Poisson process, integral representation of the fractional Poisson process, Mandelbrot--van Ness kernel, Molchan--Golosov kernel, integral with respect to the fractional Poisson process, Bichteler--Jacod inequality
Received by editor(s): February 26, 2013
Published electronically: January 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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