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Integration questions related to fractional Brownian motion. (English) Zbl 0970.60058

The authors study integration with respect to fractional Brownian motion. Their aim is to characterize the linear span \(H^{B_H}\) of the fractional Brownian motion \(B_H(t)\), \(t\in\mathbb{R}\), \(H\in(0,1)\), in terms of some function space. This means that they study integration of deterministic functions. For \(H\in (0,1/2)\) it is possible to give a complete characterization of the space \(H^{B_H}\) as follows: Let \[ \Lambda^H=\{f: \exists\varphi\in L^2 (\mathbb{R}) \text{ s.t. }f=I_-^{1/2-H} \varphi\}, \] where \(I_-^\alpha\) is the fractional integral operator of order \(\alpha\). For \(f,g\in \Lambda^H\) put \[ (f,g)_{\Lambda^H}= c_H(\varphi_f, \varphi_g)_{L^2(\mathbb{R})}. \] Then the space \(\Lambda_H\) is isometric to \(H^{B_H}\). For \(H\in(1/2,1)\) the authors show that it is not possible to obtain such a characterization in terms of a function space.
The authors also study related problems in the spectral domain using the spectral representation of fractional Brownian motion in terms of a complex Gaussian measure. They consider the following function space \[ \widetilde \Lambda^H= \Bigl\{f:f\in L^2(\mathbb{R}), \int_\mathbb{R}\bigl |\widehat f(x)\bigr |^2|x|^{1-2H} dx<\infty \Bigr\}. \] They show that the elementary functions are dense in \(\widetilde\Lambda^H\), but this space is not complete, unless \(H=1/2\). In addition, they show that \(\widetilde\Lambda^H \subset\Lambda^H\), the inclusion is strict, when \(H\neq 1/2\), and where the space \(\Lambda^H\) for \(H\in (1/2,1)\) is defined as \[ \Lambda^H=\Bigl\{f: \int_\mathbb{R} \bigl(I_-^{H-1/2}(f)(s) \bigr)^2ds <\infty\Bigr\}. \]

MSC:

60H05 Stochastic integrals
60G18 Self-similar stochastic processes
26A33 Fractional derivatives and integrals
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