# Itō isometry

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 ${\displaystyle W:[0,T]\times \Omega \to \mathbb {R} }$ denote the canonical real-valued Wiener process defined up to time ${\displaystyle T>0}$, and let ${\displaystyle X:[0,T]\times \Omega \to \mathbb {R} }$ be a stochastic process that is adapted to the natural filtration ${\displaystyle {\mathcal {F}}_{*}^{W}}$ of the Wiener process. Then

${\displaystyle \mathbb {E} \left[\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\right)^{2}\right]=\mathbb {E} \left[\int _{0}^{T}X_{t}^{2}\,\mathrm {d} t\right],}$

where ${\displaystyle \mathbb {E} }$ denotes expectation with respect to classical Wiener measure ${\displaystyle \gamma }$. 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

${\displaystyle (X,Y)_{L^{2}(W)}:=\mathbb {E} \left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\int _{0}^{T}Y_{t}\,\mathrm {d} W_{t}\right)=\int _{\Omega }\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\int _{0}^{T}Y_{t}\,\mathrm {d} W_{t}\right)\,\mathrm {d} \gamma (\omega )}$

and

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

## References

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