Itô isometry explained

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 : [ 0 , T] x \Omega \to R denote the canonical real-valued Wiener process defined up to time T > 0 , and let X : [ 0 , T] x \Omega \to R be a stochastic process that is adapted to the natural filtration l{F} ^{W} _{*} of the Wiener process. Then\operatorname{E} \left[ \left( \int ^{T} _{0} X _{t} d W _{t} \right) ^{2} \right] = \operatorname{E} \left[ \int ^{T} _{0} X ^{2} _{t} d t \right],

where

\operatorname{E} denotes expectation with respect to classical Wiener measure.

In other words, the Itô integral, as a function from the space

L ^{2} _{ad} ([ 0 ,T] x \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{align} ( X, Y ) _{L ^{2} _{ad} ([ 0 ,T] x \Omega)} & : = \operatorname{E} \left( \int ^{T} _{0} X _{t} Y _{t} d t \right) \end{align}

and

( A, B ) _{L ^{2} (\Omega)} : = \operatorname{E} ( A B ) . As a consequence, the Itô integral respects these inner products as well, i.e. we can write\operatorname{E} \left[ \left( \int ^{T} _{0} X _{t} d W _{t} \right) \left( \int ^{T} _{0} Y _{t} d W _{t} \right) \right] = \operatorname{E} \left[ \int ^{T} _{0} X _{t} Y _{t} d t \right] for X, Y \in L ^{2} _{ad} ([ 0 ,T] x \Omega) .

References

• Book: Bernt Øksendal

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