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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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