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

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

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

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

and

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

References

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