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

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  • Essaky, El Hassan
  • Nualart, David

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.

Suggested Citation

  • Essaky, El Hassan & Nualart, David, 2015. "On the 1H-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter H<12," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4117-4141.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:11:p:4117-4141
    DOI: 10.1016/j.spa.2015.06.001
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    References listed on IDEAS

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    1. Guerra, João M.E. & Nualart, David, 2005. "The 1/H-variation of the divergence integral with respect to the fractional Brownian motion for H>1/2 and fractional Bessel processes," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 91-115, January.
    2. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
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    1. Mishura, Yuliya & Ralchenko, Kostiantyn, 2024. "Fractional diffusion Bessel processes with Hurst index H∈(0,12)," Statistics & Probability Letters, Elsevier, vol. 206(C).

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