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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 $B_H(t)$, $t\in\bbfR$, $H\in(0,1)$, 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 $$\Lambda^H=\{f: \exists\varphi\in L^2 (\bbfR) \text{ s.t. }f=I_-^{1/2-H} \varphi\},$$ where $I_-^\alpha$ is the fractional integral operator of order $\alpha$. For $f,g\in \Lambda^H$ put $$(f,g)_{\Lambda^H}= c_H(\varphi_f, \varphi_g)_{L^2(\bbfR)}.$$ 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 $$\widetilde \Lambda^H= \Bigl\{f:f\in L^2(\bbfR), \int_\bbfR\bigl |\widehat f(x)\bigr |^2|x|^{1-2H} dx<\infty \Bigr\}.$$ 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 $H\ne 1/2$, and where the space $\Lambda^H$ for $H\in (1/2,1)$ is defined as $$\Lambda^H=\Bigl\{f: \int_\bbfR \bigl(I_-^{H-1/2}(f)(s) \bigr)^2ds <\infty\Bigr\}.$$

60H05Stochastic integrals
60G18Self-similar processes
26A33Fractional derivatives and integrals (real functions)
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