Some Processes Associated with Fractional Bessel Processes

Let $B=\{(B_t^1,\mathrm{.}\mathrm{.}\mathrm{.},B_t^d),t\ge 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=% \sqrt{(B_{t}^{1})^{2}+… +(B_{t}^{d})^{2}}$ be the fractional Bessel process. It^{o}'s formula for the fractional Brownian motion leads to the equation $ R_{t}=\sum_{i=1}^{d}\int_{0}^{t}\frac{B_{s}^{i}}{R_{s}}% dB_{s}^{i}+H(d-1)\int_{0}^{t}\frac{s^{2H-1}}{R_{s}}ds . $ In the Brownian motion case ($H=1\mathrm{/}2$), $X_{t}=\sum_{i=1}^{d}\int_{0}^{t} frac{B_{s}^{i}}{% R_{s}}dB_{s}^{i}$isaBrownianmotion\mathrm{.}Inthispaperitisshownthat$X_{t}$is\underset{‾}{\text{not}}afractionalBrownianmotionif$H\not=1/2$. We will study some other properties of this stochastic process as well.