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By Darrell J Kemp; Ronald L Rutowski; M Webster

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Valkeila, and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli, 5 (4) (1999), 571–587. [23] V. S. Taqqu, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields, 118 (2) (2000), 251–291. [24] J. Pr¨ uss, Evolutionary Integral Equations and Applications. Birkh¨ auser Verlag, Basel, 1993. Monographs in Mathematics, 87. [25] S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives.

2. , p1 + p1 = 1, and similarly for q. Then the quantity sμ 2−γ,q ,p ,t is bounded by sμ 2 −γ,q ,p ,t ≤ O(1) μ−(1−2γq )/(ρq ) + t(1−2(ρ+γ)q )/q μ−2 + t(3−2(1+γ)p −2ρ)/p μ−2/p + μ(2γp +2p −3)/ρp if 2(γ + ρ)q = 1 and γp + p = 3 2 − ρ. 1. 1) with respect to some kernels of the classes proposed in the introduction. 46 S. Bonaccorsi The first example is a simple perturbation of a fractional convolution operator, provided by t (t − s)(2t − s) Kϑ (t, s) = ϑ , 0 < s < t < T; this kernel belongs to Eϑ,∞,∞,t and its norm remains bounded for t → ∞.

15) s 3. 10). Again, we employ the fact that we may consider each class of work separately. We consider, for any k, the scalar stochastic convolution process t bk (t) = 0 t sμk (t − r) dB (k) (r) = 0 s∗μk (t, r) dβ (k) (r), t ∈ [0, T ]. 1. Assume that the kernel K(t, s) belongs to Eγ,q,p,t for a set of ad1 1 , p > 1−γ . Then it holds that missible parameters 0 ≤ γ < 12 , q > 1−2γ s∗μ (t, ·) 2 L2 (0,t) ≤ O(1) sμ 2 −γ,q ,p ,t K 2 γ,q,p,t . Proof. Our goal is to evaluate the L2 -norm of the function s∗μ .

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