# Does Itō isometry have different versions?

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 $$( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega).$$

Wikipedia claims the reference for the above is Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. However, I didn't find things like "$\mathbb{E}$ denotes expectation with respect to classical Wiener measure $\gamma$" in the book.

My understanding of Itō isometry is that given fixed $T$, $\left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2}$ and $\int_{0}^{T} X_{t}^{2} \, \mathrm{d} t$ are both random variables not stochastic processes, and $\mathbb{E}$ denotes expectation with respect to the probability measure on the underlying probability space $\Omega$. Is my understanding correct?

Why does Wiki's Itō isometry treat $\left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2}$ and $\int_{0}^{T} X_{t}^{2} \, \mathrm{d} t$ as stochastic processes, and $\mathbb{E}$ as expectation with respect to classical Wiener measure $\gamma$ which is a measure on the functional space $\mathbb{R}^{[0,T]}$ induced by the Wiener process $W$? I am not sure if Wiki is consistent with itself, because in the last two formulas for the inner products, the integrals are wrt the classical Wiener measure over the underlying probability space $\Omega$ instead of over the functional space. Or do I misunderstand Wiki?

Is Wiki's Itō isometry a different version from the the one that I understand? If yes, is there some reference for Wiki's version and relation between the two?

Thanks and regards!

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1. Why do you think that Wiki treats them as stochastic processes? 2. What is the underlying probability space $\Omega$? – Ilya Nov 27 '11 at 11:44
@Ilya: Thanks! 1.Because the classical Wiener measure is defined on the functional space of functions from the index set to the state space. I am not sure if Wiki is consistent with itself, because in the last two formulas for the inner products, the integrals are wrt the classical Wiener measure over the underlying probability space $\Omega$ instead of over the functional space. 2. The underlying probability space $\Omega$ is the one for the Wiener process $W$, the process $X$ and the filtration $\mathcal{F}_{*}^{W}$. Hope I make myself clearer. – Tim Nov 27 '11 at 13:25
@Tim : Hi I'm not sure to understand your first question. What does the fact that you see $(\int_{0}^{T} X_{t} dW_{t} )^{2}$ and $\int_{0}^{T} X_{t}^{2} dt$ as random variable or as stochastic processes has to do with Itô's Isometry, both are $\mathcal{F}_T$-measurable a sigma field built upon history of the process ? – TheBridge Nov 28 '11 at 16:54
@TheBridge: The two quantities are random variables for fixed time T and process X, and are stochastic processes for varying time T and fixed process X. Wiki's Ito isometry seems to treat them as processes i.e. with varying T, while my understanding of Ito isometry is for random variables i.e. with fixed T. The next question is, what probability space is the expectation E taken relative to, the functional space of functions from index to state i.e. the codom of processes as Wiki says, or the domain $\Omega$ of the random variables as I understand? – Tim Nov 28 '11 at 17:09

Since W is the canonical Wiener process you have right from the definition that $\Omega$ is equal to the space of functions from the index set to the state space (usually denoted by $C_0$), i.e.

$$(\Omega,\mathcal{A},\mathbb{P})=(C_0,\mathcal{B}(C_0),\mu)$$

where $\mu$ denotes the Wiener measure. So Wiki is consistent with itself.

Moreover, Wiki does not treat $\left( \int_0^T X_t \, dW_t \right)^2$ and $\int_0^T X_t^2 \, dt$ as stochastic processes, but as random variables

$$C_0 \ni \omega \mapsto \left( \int_0^T X_t \, dW_t \right)^2(\omega)$$

resp.

$$C_0 \ni \omega \mapsto \left(\int_0^T X_t^2 \, dt \right)(\omega)$$

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