Some Processes Associated with Fractional Bessel Processes

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Let B={(Bₜ¹,..., Bₜᵈ), t≥ 0} be a d-dimensional fractional Brownian motion with Hurst parameter H and let Rₜ=% √(Bₜ¹)²+... +(Bₜᵈ)² be the fractional Bessel process. Itô's formula for the fractional Brownian motion leads to the equation Rₜ=∑ᵢ₌₁ᵈ∫₀ᵗBₛⁱRₛ% dBₛⁱ+H(d-1)∫₀ᵗs²ʰ⁻¹Rₛds. In the Brownian motion case (H=1/2), Xₜ=∑ᵢ₌₁ᵈ∫₀ᵗ fracBₛⁱ% RₛdBₛⁱ is a Brownian motion. In this paper it is shown that Xₜ is not a fractional Brownian motion if H=1/2. We will study some other properties of this stochastic process as well.