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By Sanz-Sole M.

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Let K = {K(s, z), (s, z) ∈ [0, T ] × Rd } be a K-valued predictable process; we assume the following condition: Hypothesis B The process K satisfies sup(s,z)∈[0,T ]×Rd E ||K(s, z)||2K < ∞. Our first purpose is to define a martingale measure with values in K obtained by integration of K. Let {ej , j ≥ 0} be a complete orthonormal system of K. Set K j (s, z) = K(s, z), ej K , (s, z) ∈ [0, T ] × Rd . According to [64], for any j ≥ 0 the process t j MtK (A) = 0 K j (s, z)M (ds, dz), t ∈ [0, T ], A ∈ Bb (Rd ), A defines a martingale measure.

S. The assumption (2) and the duality formula yields n E ϕ(F )δ (γ −1 )i,l DF l l=1 n E D(ϕ(F )), (γ −1 )i,l DF l = H l=1 = E ∂i ϕ(F ) . 4) is proved. Notice that by assumption Hi (F, 1) ∈ L2 (Ω). 2 part 1) yields the existence of density. 10 one can give sufficient conditions ensuring the validity of the above assumption 2 and an alternative form of the random variables Hi (F, 1), as follows. s. and for any i, j = 1, · · · , n, F j ∈ Dom L, (γ −1 )i,j ∈ D1,2 , (γ −1 )i,j DF j ∈ L2 (Ω, H), (γ −1 )i,j δ(DF j ) ∈ L2 (Ω), D(γ −1 )i,j , DF j H ∈ L2 (Ω).

We have chosen here one of these applications -the most known- to a problem in option pricing. The choice was made of the basis of its theoretical interest. 44 In fact it gives further insight to Itˆo’s result on representation of square integrable random variables. We have followed the lecture notes [49]. 1 are from Nualart and Zakai (see [45]). 2 appears in [23]. 1 Let W be a white noise based on (A, A, m). Consider a random variable F ∈ DomDk and G ∈ A. Prove that Dtk E(F/FG ) = E(Dtk F/FG )1 1Gk (t).

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