By Yasui Y.

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Let u ∈ Lp (Ω, Lq (R, R, dr), P) and let Let p ∈ (1, ∞[ and q ∈ ( 1/2+H {uk }k∈N be a sequence in Dom∗ δ W lim uk = u k→∞ H ∩ Lp (Ω, Lq (R, R, dr), P) such that in Lp Ω, Lq (R, R, dr), P . If there exist a pˆ ∈ (1, ∞] and an X ∈ Lpˆ(Ω, R, P) such that lim δ(uk ) = X in Lpˆ(Ω, R, P), k→∞ H H then u ∈ Dom∗ δ W , and δ W (u) = X. 5 Link with white noise theory In order to describe the approach used by Biagini, Øksendal, Sulem, and Wallner in [BØSW04], we present a summary of classical white noise theory.

To I1,− H K is close to a Liouville operator, see [SKM93]. The operator I1,− Stochastic Calculus with Respect to FBM 35 Lemma 5. For H ∈ ]0, 1[, one has the following identification: for suitable a H H− 1 1 1 K I1,− (a)(s) = cH s 2 −H I1,− 2 uH− 2 a(u) (s), s ∈ [0, 1]. Here according to [SKM93] for α ∈]0, 1[ and f ∈ Lploc (R, R, dx), s < 1, α (f )(s) = I1,− 1 Γ (α) 1 f (u)(u − s)α−1 du s and for suitable f −α (f )(s) = I1,− f (s) 1 −α Γ (1 − α) (1 − s)α 1 s f (u) − f (s) du . (u − s)α+1 Proof.

We say H that u ∈ Dom∗ δ W whenever there exists in ∪p>1 Lp (Ω, R, P) a random variH able δ W (u) such that for all n ∈ N∗ and φ ∈ E verifying φ L2 (R) = 1, the following conditions are satisfied: 1. ) ∈ L1 ([0, 1]), 2. E [u. Hn−1 (B(φ))] (I1,− H 1 2 K H −1,∗ 3. CH E [u(t) Hn−1 (B(φ))] (I1,− ) (φ)(t) dt = E δ W (u) Hn B(φ) , 0 H K where I1,− −1,∗ H K is the adjoint of I1,− 2 CH = −1 in L2 ([0, 1], R, dr) and Γ (H + 1/2)2 ∞ 0 (1 + s)H−1/2 − sH−1/2 H 2 ds + 1/(2H) . H Observe that if u ∈ Dom∗ δ W , then δ W (u) is unique, and the mapping H δ : Dom∗ δ W → ∪p>1 Lp (Ω, R, P) is linear.

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