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Properties of integrals with respect to fractional Poisson processes with compact kernels. (English) Zbl 1326.60080

Theory Probab. Math. Stat. 89, 143-152 (2014); translation from Teor. Jmovirn. Mat. Stat. 89, 130–139 (2013).
Summary: 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.

MSC:

60H05 Stochastic integrals
60G22 Fractional processes, including fractional Brownian motion
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G51 Processes with independent increments; Lévy processes
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