Let (left(B_{t} ight)_{t geqslant 0}) be a (mathrm{BM}^{1}), denote by (mathcal{H}_{T}^{2}) the space used in Lemma 19.10

Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\), denote by \(\mathcal{H}_{T}^{2}\) the space used in Lemma 19.10 and pick \(\xi_{1}, \ldots, \xi_{n} \in \mathbb{R}\) and \(0=t_{0}]\) is contained in \(\mathcal{H}_{T}^{2} \oplus i \mathcal{H}_{T}^{2}\).

Data From  Lemma 19.10

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19.10 Lemma. The space H is a closed linear subspace of L(Q, FT, P). Proof. His obviously a linear space and, by Theorem 15.15, it is contained in L(Q, FT, P). We show that it is a closed subspace. Let (M)n1 CH, T M = x + [x dBs, 0 be an L(P) Cauchy sequence. Thus, it is an L(P) Cauchy sequence, and we see from E[MM] x - xm = that (x") is a Cauchy sequence in R; denote its limit by x = lim cox". This means that M-x=x dB, is again a Cauchy sequence in L(P). Using It's Isometry (15.22) we see that E= [(M? x") (MF?" -x))}] = [([{a ) = [([CX] - X75) dB.;)"] = [[|X3 X37; 5] . E (X-X) EIX-X2 ds i.e. (X)ST is a Cauchy sequence in L(ATP). Because of the completeness of L(ATP) the sequence converges to some process (XS)ser and, for some subsequence, we get lim X() (w) = Xs(w) for ATP almost all (s, w) and in I (+ &P). Since pointwise limits preserve measurability, we have (Xs)s