Working towards the Ito Isometry

We have defined the stochastic integral (H\cdot A)_t for a finite variation, continuous adapted process A, and H a previsible process. We want to extend this to integrals with respect to continuous local martingales. Analogous to defining the Lebesgue integral by extending linearly from indicator functions, we define integrals of simple processes with respect to martingales. A simple process is a disjointly (time-)supported linear combination of the canonical previsible process Z_s1_{(s,t]}. It will be convenient to demand that the variables of integration are in \mathcal{M}^2. Then the space \mathcal{M}^2 is preserved under this integral operation, that is H\cdot M\in\mathcal{M}^2, and in particular

\mathbb{E}(H\cdot M)_\infty^2\leq ||H||_\infty^2\mathbb{E}(M_\infty-M_0)^2

By Doob’s L^2 inequality, X\in\mathcal{M}^2\Rightarrow ||\sup_t|X_t|||_2<\infty, so it makes sense to define \mathcal{C}^2 as the set of cadlag adapted processes s.t. |||X|||:=||\sup_t|X_t|||<\infty. Of particular interest is that \mathcal{C}^2 is complete, by finding limits for subsequences and lifting to the whole sequence by Fatou.

We aim to construct integrals over \mathcal{M}_{c,loc} by taking limits of integrals over \mathcal{M}_c^2. For this space, we have that M^2-[M] is not just \in\mathcal{M}_{c,loc} but in fact a UI true martingale. Then, as a natural extension of the orthogonality trick (remembering that M^2-[M]\in\mathcal{M}_{c,loc}), we obtain

\mathbb{E}[M]_\infty=\mathbb{E}(M_\infty-M_0)^2

We then define for M\in\mathcal{M}_c^2 the norm ||\cdot||_M by ||H||_M^2:=\mathbb{E}\left(\int_0^tH_s^2d[M]_s\right) with L^2(M) the space of H s.t. that this is finite. Then for H\in \mathcal{S}, we have the Ito isometry:

I: (L^2(M),||\cdot||_M)\rightarrow(M_c^2,||\cdot||) with I(H)=H\cdot M

Now, we also have S dense in L^2(M). Why? We know linear combinations of 1_A, A\in\mathcal{P} are dense, so it STP A\in\mathcal{P}\Rightarrow A\in\bar{S}. Use a typical Dynkin’s lemma argument on \mathcal{A}=\{A\in\mathcal{P}:1_A\in \bar{S}\}, which contains \pi-system generating \mathcal{P}. So extend to L^2(M) generally. We recover the original motivating result that [H\cdot M]=H^2\cdot[M]. Take a sequence of stopping times reducing both H and M to force boundedness of integrand, and M\in\mathcal{M}_c^2. Stopping the integral is equivalent to stopping the integrand, and checking limits in stopping times allows us to lift the Ito Isometry result to this one about quadratic variations.

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