View source for Itô isometry

In [[Mathematics|mathematics]], the '''Itô isometry''', named after Kiyoshi Itô, is a crucial fact about [[Itô calculus|Itô stochastic integrals]]. One of its main applications is to enable the computation of [[Variance|variance]]s for random variables that are given as Itô integrals.
Let <math>W : [0, T] \times \Omega \to \mathbb{R}</math> denote the canonical real-valued [[Wiener process]] defined up to time <math>T > 0</math>, and let <math>X : [0, T] \times \Omega \to \mathbb{R}</math> be a [[Stochastic process|stochastic process]] that is [[Adapted process|adapted]] to the [[Natural filtration|natural filtration]] <math>\mathcal{F}_{*}^{W}</math> of the Wiener process.{{what|reason= Doesn't X have to be progressively measurable for the Itô integral even to be defined?|date=October 2023}} Then

:<math>\operatorname{E} \left[ \left( \int_0^T X_t \, \mathrm{d} W_t \right)^2 \right] = \operatorname{E} \left[ \int_0^T X_t^2 \, \mathrm{d} t \right],</math>

where <math>\operatorname{E}</math> denotes [[Expected value|expectation]] with respect to classical Wiener measure. 

In other words, the Itô integral, as a function from the space <math>L^2_{\mathrm{ad}} ([0,T] \times \Omega)</math> of square-integrable [[Adapted process|adapted processes]] to the space <math>L^2 (\Omega)</math> of square-integrable random variables, is an [[Isometry|isometry]] of [[Normed vector space|normed vector space]]s with respect to the norms induced by the inner products

:<math>
\begin{align}
( X, Y )_{L^2_{\mathrm{ad}} ([0,T] \times \Omega)} & := \operatorname{E} \left( \int_0^T X_t \, Y_t \, \mathrm{d} t \right)
\end{align}
</math>

and

:<math>( A, B )_{L^2 (\Omega)} := \operatorname{E} ( A B ) .</math>
As a consequence, the Itô integral respects these inner products as well, i.e. we can write
:<math>\operatorname{E} \left[ \left( \int_0^T X_t \, \mathrm{d} W_t \right) \left( \int_0^T Y_t \, \mathrm{d} W_t \right) \right] = \operatorname{E} \left[ \int_0^T X_t Y_t \, \mathrm{d} t \right]</math>
for <math>X, Y \in L^2_{\mathrm{ad}} ([0,T] \times \Omega)</math> .

== References ==
* {{cite book | author=Øksendal, Bernt K. |  title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer, Berlin | year=2003 | isbn=3-540-04758-1}}

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[[Category:Stochastic calculus]]

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