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 W : , T\times \Omega \to \mathbb denote the canonical real-valued Wiener process defined up to time $T > 0$, and let X : , T\times \Omega \to \mathbb be a stochastic process that is adapted to the natural filtration $\mathcal_^$ of the Wiener process. Then

:\operatorname \left \left( \int_0^T X_t \, \mathrm W_t \right)^2 \right= \operatorname \left \int_0^T X_t^2 \, \mathrm t \right

where $\operatorname$ denotes expectation with respect to classical Wiener measure.

In other words, the Itô integral, as a function from the space L^2_ ( ,T\times \Omega) of square-integrable adapted processes to the space $L^2 (\Omega)$ of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products

:$\begin ( X, Y )_ & := \operatorname \left( \int_0^T X_t \, Y_t \, \mathrm t \right) \end$

and

:$( A, B )_ := \operatorname ( A B ) .$
As a consequence, the Itô integral respects these inner products as well, i.e. we can write
:\operatorname \left \left( \int_0^T X_t \, \mathrm W_t \right) \left( \int_0^T Y_t \, \mathrm W_t \right) \right= \operatorname \left \int_0^T X_t Y_t \, \mathrm t \right/math>
for X, Y \in L^2_ ( ,T\times \Omega) .

References

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{{DEFAULTSORT:Ito Isometry
Stochastic calculus