What are the norms in Ito isometry?

Question

Itō isometry from Wikipedia:

Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to \mathbb{R}$ be a stochastic process that is adapted to the natural filtration $\mathcal{F}_{*}^{W}$ of the Wiener process. Then $$ \mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \, \mathrm{d} t \right], $$ where $\mathbb{E}$ denotes expectation with respect to classical Wiener measure $\gamma$. In other words, the Itō stochastic integral, as a function, is an isometry of normed vector spaces with respect to the norms induced by the inner products $$ ( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) = \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma (\omega) $$ and $$ ( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega). $$

I was wondering what the two normed spaces are and what their norms are, so that the two normed spaces are isometric wrt their norms?

Thanks and regards!

Answer 1

Let $(W_t)_{t \geq 0}$ a Wiener process on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. Denote by $X \bullet W_T$ the stochastic integral $$X \bullet W_T := \int_0^T X_t \, dW_t$$ Then Itô Isometry states $$\|X \bullet W_T\|_{L^2(\mathbb{P})}^2 = \|X\|_{L^2(\lambda|_{[0,T]} \times \mathbb{P})}^2$$ where $\lambda|_{[0,T]}$ denotes the Lebesgue measure on $[0,T]$.

This means that the mapping $L^2(\lambda|_{[0,T]} \times \mathbb{P}) \ni X \mapsto X \bullet W_T \in L^2(\mathbb{P})$ is an isometry between the normed space

• $$L^2(\mathbb{P}) := \left\{f: \Omega \to \mathbb{R}; f \, \text{measurable}, \int_{\Omega} f(\omega)^2 \, d\mathbb{P}(\omega)< \infty\right\}$$ endowed with the norm $$\|f\|_{L^2(\mathbb{P})} := \left( \int_{\Omega} f(\omega)^2 \, d\mathbb{P}(\omega) \right)^{\frac{1}{2}} $$

and the normed space

• $$L^2(\lambda|_{[0,T]} \times \mathbb{P}):= \left\{ f:[0,T] \times \Omega \to \mathbb{R}; f \, \text{measurable}, \int_{\Omega} \int_0^T f(t,\omega)^2 \, dt \, d\mathbb{P}(\omega)< \infty\right\}$$ endowed with the norm $$\|f\|_{L^2(\lambda|_{[0,T]} \times \mathbb{P})} := \left(\int_{\Omega} \int_0^T f(t,\omega)^2 \, dt \, d\mathbb{P}(\omega) \right)^{\frac{1}{2}}$$

In the Wikipedia article, they consider $W$ as canonical Wiener process, i.e. $$\Omega := C_{(0)} := \{w:[0,\infty) \to \mathbb{R}; w \, \text{continuous}, w(0)=0\}$$ and $\mathbb{P}=\gamma$ is given by the Wiener measure.