Itô isometry

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

Let denote the canonical real-valued Wiener process defined up to time , and let be a stochastic process that is adapted to the natural filtration of the Wiener process. Then

where denotes expectation with respect to classical Wiener measure. In other words, the Itô integral, as a function, is an isometry of normed vector spaces with respect to the norms induced by the inner products

and

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