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

$\mathrm{E}\left[{\left(\underset{0}{\overset{T}{\int }}{X}_t\mathrm{d}{W}_t\right)}^2\right]=\mathrm{E}\left[\underset{0}{\overset{T}{\int }}{X}_t^2\mathrm{d}t\right],$

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

In other words, the Itô integral, as a function from the space $L_{\mathrm{a}\mathrm{d}}^2([0,T]\times \mathrm{\Omega })$ of square-integrable adapted processes to the space $L^2(\mathrm{\Omega })$ of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products

$\begin{array}{rl}(X,Y{)}_{L_{\mathrm{a}\mathrm{d}}^2([0,T]\times \mathrm{\Omega })}& :=\mathrm{E}\left(\underset{0}{\overset{T}{\int }}{X}_t{Y}_t\mathrm{d}t\right)\end{array}$

and

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

As a consequence, the Itô integral respects these inner products as well, i.e. we can write

$\mathrm{E}\left[\left(\underset{0}{\overset{T}{\int }}{X}_t\mathrm{d}{W}_t\right)\left(\underset{0}{\overset{T}{\int }}{Y}_t\mathrm{d}{W}_t\right)\right]=\mathrm{E}\left[\underset{0}{\overset{T}{\int }}{X}_t{Y}_t\mathrm{d}t\right]$

for $X,Y\in L_{\mathrm{a}\mathrm{d}}^2([0,T]\times \mathrm{\Omega })$ .

References

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