Some Processes Associated with Fractional Bessel Processes

Abstract

Let $B = { (B_t^{1}, ..., B_t^{d} ),t \geq 0}$ be a d-dimensional fractional Brownian motion with Hurst parameter H and let $R_{t} = \sqrt {(B_t^1 )^2 + ... + (B_t^{d} )^{2} }$ be the fractional Bessel process. Itô’s formula for the fractional Brownian motion leads to the equation $R_t = \sum_{i = 1}^d ,\int_0^{t} \frac{B_s^{i} }{R_{s} }\ {d} B_s^i + H(d -1)\int_0^{t} \frac{s^{2H - 1}} {R_s }\ {d} s$ . In the Brownian motion case $(H=1/2), X_t = \sum\nolimits_{i = 1}^d {\int_0^t {\frac{{B_s^i }} {{R_s }}} } \d B_s^i$ is a Brownian motion. In this paper it is shown that X_t is not an ${\cal F}^{B}$ -fractional Brownian motion if H ≠ 1/2. We will study some other properties of this stochastic process as well.