Itô isometry
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 random variables that are given as Itô integrals. Let formula_1 denote the canonical real-valued Wiener process defined up to time formula_2, and let formula_3 be a stochastic process that is adapted to the natural filtration formula_4 of the Wiener process. Then where formula_6 denotes expectation with respect to classical Wiener measure. In other words, the Itô integral, as a function from the space formula_7 of square-integrable  adapted processes to the space formula_8 of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products and As a consequence, the Itô integral respects these inner products as well, i.e. we can write for formula_12 .