Some Processes Associated with Fractional Bessel Processes

Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=$ be the fractional Bessel process. It\^{o}'s formula for the fractional Brownian motion leads to the equation $R_{t}=\sum_{i=1}^{d}\int_{0}^{t}\frac{B_{s}^{i}}{R_{s}}$ In the Brownian motion case ($H=1/2$), X_{t}=\sum_{i=1}^{d}\int_{0}^{t} frac{B_{s}^{i}}{% R_{s}}dB_{s}^{i} is a Brownian motion. In this paper it is shown that $X_{t}$ is \underbar{not} a fractional Brownian motion if $H\not=1/2$. We will study some other properties of this stochastic process as well.