Some Processes Associated with Fractional Bessel Processes

Y. Hu () and D. Nualart ()
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Y. Hu: University of Kansas
D. Nualart: Universitat de Barcelona

Journal of Theoretical Probability, 2005, vol. 18, issue 2, 377-397

Abstract: 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 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.

Keywords: Fractional Brownian motion; fractional Bessel processes; stochastic integral; Malliavin derivative; chaos expansion (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/s10959-005-3508-7

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