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
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
![{\displaystyle \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],}](/index.php?title=Special:MathShowImage&hash=89cd5efa704c5c34b62549c005b3becd&mode=mathml)
where
denotes expectation with respect to classical Wiener measure
. 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

and

References
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