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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 random variables that are given as Itô integrals.

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.[clarification needed] Then

where denotes expectation with respect to classical Wiener measure.

In other words, the Itô integral, as a function from the space of square-integrable adapted processes to the space of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products


As a consequence, the Itô integral respects these inner products as well, i.e. we can write

for .


  • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1.