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Are classes of deterministic integrands for fractional Brownian motion on an interval complete? (English) Zbl 1003.60055
Let \(B_\kappa\) be a fractional Brownian motion (fBM) with Hurst-parameter \(H=\kappa+1/2\in(0,1)\) and \(a>0\) be a fixed real number. The authors prove that for \(\kappa\in(-1/2,0)\) the inner product space \[ \Lambda_a^\kappa=\{f:\exists\varphi_f\;\text{such\;that} f(u)=u^{-\kappa}(I_{a-}^{-\kappa}s^\kappa\varphi_f(s))(u)\} \] is complete, where \(I_{a-}^\alpha\) is the right-sided fractional integral of order \(\alpha>0\) on \([0,a]\). Therefore the (stochastic) integral \[ {\mathcal I}_a^\kappa (f)=\int_0^a f(u) dB_\kappa(u) =\text{const}(\kappa)\int^a_0 u^{-\kappa}(I_{a-}^\kappa s^\kappa f(s))(u) dB_0(u) \] is an isometry into the space \(\overline{\text{sp}}_{[0,a]}(B_\kappa)\) which is the \(L^2\)-closure of all linear combinations of the increments of \(B_\kappa\) on \([0,a]\). On the other hand, it is shown that for \(\kappa\in(0,1/2)\) the inner product space \(\Lambda_a^\kappa\) introduced for example by M. L. Kleptsyna, M.-C. Roubaud and A. Le Breton [in: Probability theory and mathematical statistics, 373-392 (1999; Zbl 0999.60038)] and the inner product space \(|\Lambda|_a^\kappa\) introduced for example by I. Norros, E. Valkeila and J. Virtamo [Bernoulli 5, No. 4, 571-587 (1999; Zbl 0955.60034)] are not complete. Thus they are only isometric to proper linear subspaces of \(\overline{\text{sp}}_{[0,a]}(B_\kappa)\), respectively. The reason of these facts is that completeness is equivalent to the solvability of \[ u^{-\kappa}(I_{a-}^\kappa s^\kappa f(s))(u)=\varphi(u) \] for every \(\varphi\in L^2[0,a]\). Yet, this is fulfilled for \(\kappa\in(-1/2,0)\), but not for \(\kappa\in(0,1/2)\). Nevertheless, the authors show that for \(\kappa\in(-1/2,1/2)\) the conditional expectation \(X=E\{B_\kappa(t)\mid B_\kappa(s),s\in[0,a]\}\) for \(0<a<t\) can still be expressed as \(X=\int_0^a f(u) dB_\kappa(u)\) for some function \(f\). The results about completeness are extended to the case \(a=\infty\), too. Moreover, a concise overview of the representation of fBM in terms of fractional integrals is given.

MSC:
60H05 Stochastic integrals
60J65 Brownian motion
60G15 Gaussian processes
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