Itō isometry

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In mathematics, the Itō isometry, named after Kiyoshi Itō, 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[ \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)


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