How can we extend the Itō isometry for elementary processes to the class of square-integrable processes?

Question

Let

• $\lambda^1$ denote the Lebesgue measure on $\mathbb R$
• $T>0$
• $I:=(0,T]$
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $(\mathcal F_t)_{t\in\overline I}$ be a filtration of $\mathcal A$
• $(W_t)_{t\in\overline I}$ be a $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$

$\Phi:\Omega\times\overline I\to\mathbb R$ is called elementary $\mathcal F$-predictable $:\Leftrightarrow$ $$\Phi_t=\sum_{i=1}^kX_i1_{(t_{i-1},\:t_i]}\;\;\;\text{for all }t\in\overline I\tag1$$ for some $k\in\mathbb N$, $\mathcal F_{t_{i-1}}$-measurable $X_i$ with $\left|X_i(\Omega)\right|\in\mathbb N$ and $0\le t_0<\cdots<t_k\le T$. Let $$\mathcal E:=\left\{\Phi:\Omega\times\overline I\to\mathbb R\mid\Phi\text{ is elementary }\mathcal F\text{-predictable}\right\}\;.$$ Let $\Phi\in\mathcal E$ with $(1)$ $\Rightarrow$ $$\int_0^T\Phi_t\:{\rm d}W_t:=\sum_{i=1}^kX_i\left(W_{t_i}-W_{t_{i-1}}\right)$$ is called Itō integral of $\Phi$ with respect to $W$ and $$\int_a^b\Phi_t\:{\rm d}W_t:=\int_0^T\Phi_t1_{(a,b]}(t)\:{\rm d}W_t$$ is called Itō integral of $\Phi$ with respect to $W$ from $a\in[0,T]$ up to $b\in[a,T]$. Let $\Phi,\Psi\in\mathcal E$ and $0\le a\le b\le T$ $\Rightarrow$ $$\operatorname E\left[\int_a^b\Phi_t\:{\rm d}W_t\int_a^b\Psi_t\:{\rm d}W_t\mid\mathcal F_a\right]=\operatorname E\left[\int_a^b\Phi_t\Psi_t\:{\rm d}t\mid\mathcal F_a\right]\tag2\;.$$ Let $$\mathcal R:=\bigcup_{F\in\mathcal F_0}F\times\left\{0\right\}\cup\bigcup_{0\le s<t\le T}\bigcup_{F\in\mathcal F_s}F\times(s,t]$$ and $$\mathcal P:=\sigma(\mathcal R)\;.$$ $\Phi:\Omega\times\overline I\to\mathbb R$ is called $\mathcal F$-predictable $:\Leftrightarrow$ $\Phi$ is $\mathcal P$-measurable. Note that $\mathcal E$ is a dense subspace of $$\mathcal I^2:=\left\{\Phi\in\mathcal L^2\left(\operatorname P\otimes\left.\lambda^1\right|_{\overline I}\right):\Phi\text{ is }\mathcal F\text{-predictable}\right\}$$ and $\mathcal I^2$ is a closed subspace of $\mathcal L^2\left(\operatorname P\otimes\left.\lambda^1\right|_{\overline I}\right)$.

Question: How can we extend $(2)$ to $\mathcal I^2$?

Let $\Phi,\Psi\in\mathcal I^2$, $0\le a\le b\le T$ and $A\in\mathcal F_a$. We need to show that $$\operatorname E\left[1_A\int_a^b\Phi_t\:{\rm d}W_t\int_a^b\Psi_t\:{\rm d}W_t\right]=\operatorname E\left[1_A\int_a^b\Phi_t\Psi_t\:{\rm d}t\right]\tag3\;.$$ Note that $1_A$ is $\mathcal F_a$-measurable and hence $$1_{A\times(a,\:b]}\in\mathcal I^2\tag4$$ with $$1_A\int_a^b\Phi_t\:{\rm d}W_t\int_a^b\Psi_t\:{\rm d}W_t=\int_a^b1_A\Phi_t\:{\rm d}W_t\int_a^b1_A\Psi_t\:{\rm d}W_t\tag5\;.$$ Moreover, note that $\mathcal I^2$ equipped with the inner product inherited from $\mathcal L^2\left(\operatorname P\otimes\left.\lambda^1\right|_{\overline I}\right)$ is a $\mathbb R$-Hilbert space.

Answer 1

Hints:

1. Because of $$\Phi_t \Psi_t = \frac{1}{4} ((\Psi_t+\Phi_t)^2-(\Psi_t-\Phi_t)^2)$$ it suffices to show $(2)$ for $\Phi_t = \Psi_t$.
2. Show that for any sequence $X_n \to X$ in $L^2$ and any measurable set $A$, it holds that $$\mathbb{E}(1_A X_n^2) \to \mathbb{E}(1_A X^2).$$
3. For given $\Psi \in \mathcal{I}^2$ choose a sequence $(\Psi^{(n)})_{n \in \mathbb{N}} \subseteq \mathcal{E}$ such that $\Psi^{(n)} \to \Psi$ in $L^2(\mathbb{P} \otimes \lambda^1|_{[0,T]})$. By the very definition of the Itô integral, we have $$\int_a^b \Psi^{(n)}_t \, dW_t \xrightarrow[L^2]{n \to \infty} \int_a^b \Psi_t \,d W_t.$$
4. Since $(2)$ holds for $\Psi^{(n)}$ we conclude $$\begin{align*} \mathbb{E} \left( 1_A \left[ \int_a^b \Psi_t \, dW_t \right]^2 \right) &= \lim_{n \to \infty} \mathbb{E} \left( 1_A \left[ \int_a^b \Psi^{(n)}_t \, dW_t \right]^2 \right) \\ &= \lim_{n \to \infty} \mathbb{E} \left( 1_A \int_a^b (\Psi^{(n)}_t)^2 \, dt \right) \\ &= \mathbb{E} \left( 1_A \int_a^b \Psi_t^2 \, dt \right)\end{align*}$$ for all $A \in \mathcal{F}_a$.