On the 1H-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter H<12

El Hassan Essaky and David Nualart

Stochastic Processes and their Applications, 2015, vol. 125, issue 11, 4117-4141

Abstract: In this paper, we study the 1H-variation of stochastic divergence integrals Xt=∫0tusδBs with respect to a fractional Brownian motion B with Hurst parameter H<12. Under suitable assumptions on the process u, we prove that the 1H-variation of X exists in L1(Ω) and is equal to eH∫0T|us|1Hds, where eH=E[|B1|1H]. In the second part of the paper, we establish an integral representation for the fractional Bessel Process ‖Bt‖, where Bt is a d-dimensional fractional Brownian motion with Hurst parameter H<12. Using a multidimensional version of the result on the 1H-variation of divergence integrals, we prove that if 2dH2>1, then the divergence integral in the integral representation of the fractional Bessel process has a 1H-variation equals to a multiple of the Lebesgue measure.

Keywords: Fractional Brownian motion; Malliavin calculus; Skorohod integral; Fractional Bessel processes (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2015.06.001

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