Journal of Theoretical Probability

, Volume 18, Issue 2, pp 377–397 | Cite as

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 \(R_{t} = \sqrt {(B_t^1 )^2 + ... + (B_t^{d} )^{2} }\) 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 Xt 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.


Fractional Brownian motion fractional Bessel processes stochastic integral Malliavin derivative chaos expansion 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Facultat de MatemàtiquesUniversitat de BarcelonaBarcelonaSpain

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