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 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 Itô stochastic integral, as a function, is an isometry of normed vector spaces with respect to the norms induced by the inner products

$\begin{align} ( X, Y )_{L^2 (W)} & := \mathbb{E} \left( \int_0^T X_t \, \mathrm{d} W_t \int_0^T Y_t \, \mathrm{d} W_t \right) \\[6pt] & = \int_\Omega \left( \int_0^T X_t \, \mathrm{d} W_t \int_0^T Y_t \, \mathrm{d} W_t \right) \, \mathrm{d} \gamma (\omega) \end{align}$

and

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