By Stewart J.

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**Additional info for Calculus: all solutions**

**Example text**

Assume that det γ −1 ∈ ∩p∈[1,∞) Lp (Ω). 7) Then, (1) det γ −1 ∈ D∞ and γ −1 ∈ D∞ (Rm × Rm ). (2) Let G ∈ D∞ . For any multiindex α ∈ {1, · · · , n}r , r ≥ 1, there exists a random variable Hα (F, G) ∈ D∞ such that for any function ϕ ∈ Cb∞ (Rn ), E (δα ϕ)(F )G = E ϕ(F )Hα (F, G) . 9) l=1 and in general, for α = (α1 , · · · , αr−1 , αr ), Hα (F, G) = Hαr (F, H(α1 ,··· ,αr−1 ) (F, G)). 10) Proof: Consider the sequence of random variables YN = (det γ + N1 )−1 , N ≥ 1 . 7) clearly yields lim YN = det γ −1 , N →∞ in Lp (Ω).

2 j This shows that E MtK (A) K = j≥0 E |MtK (A)|2 < ∞, due to Hypothesis B. Clearly, the process {MtK (A), t ∈ [0, T ], A ∈ Bb (Rd )} defines a worthy Kvalued martingale measure and by construction we have that MtK (A), ej K = j MtK (A). By the previous computations E MtK (A) 2 K t = E 0 j≥0 ds||1 1A (·)K j (s, ·)||2H , where we have denoted by “·” the H−variable. Our next goal is to introduce stochastic integration with respect to M K , allowing the integrands to take values on some subset of the space of Schwartz distributions.

Let t → S(t) be a deterministic function with values in the space of non-negative distributions with rapid decrease, such that T µ(dξ)|FS(t)(ξ)|2 < ∞. dt 0 Rd Then S belongs to P0,K and E ||(S · M K )t ||2K = t ds Rd 0 57 2 µK s (dξ)|FS(s)(ξ)| . 8) Rd p µ(dξ)|FS(s)(ξ)|2 ) 2 −1 , t ∈ [0, T ]. Proof: Let ψ be a non-negative function in C ∞ (Rd ) with support contained in the unit ball of Rd and such that Rd ψ(x)dx = 1. Set ψn (x) = nd ψ(nx), n ≥ 1. Define Sn (t) = ψn ∗ S(t). Clearly, Sn (t) ∈ S(Rd ) for any n ≥ 1, t ∈ [0, T ] and Sn (t) ≥ 0.