Itō isometry

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In mathematics, the Itō isometry is a crucial fact about Itō stochastic integrals. One of its main applications is to enable the computation of variances for stochastic processes.

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

$\mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} = \mathbb{E} \left( \int_{0}^{T} X_{t}^{2} \, \mathrm{d} t \right),$

where $\mathbb{E}$ denotes expectation with respect to classical Wiener measure $γ$ . In other words, the Itō stochastic integral, as a function

Itō integrable processes $L^{2} (W) \subset L^{2} ([0, T] \times \Omega, \mathcal{B}([0, T]) \otimes \mathcal{B}(\Omega), \lambda \otimes \gamma; \mathbb{R}) \to L^{2} (\Omega, \mathcal{B}(\Omega), \gamma; \mathbb{R})$

is an isometry of normed vector spaces with respect to the norms induced by the inner products

$( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) = \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma (\omega)$