Some Processes Associated with Fractional Bessel Processes

Details Some Processes Associated with Fractional Bessel Processes Some Processes Associated with Fractional Bessel Processes

2004-02-02

arxiv.org/abs/math/0402019v1

Mathematics

Probability

Scientific paper

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. 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 ($H=1/2$), $X_{t}=\sum_{i=1}^{d}\int_{0}^{t} frac{B_{s}^{i}}{% R_{s}}dB_{s}^{i}$ 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.

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