Some Processes Associated with Fractional Bessel Processes

Abstract

Let $B=\{(B_t^1,...,B_t^d),t\ge 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_t=$ be the fractional Bessel process. Itô'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 $H\ne 1/2$. We will study some other properties of this stochastic process as well.