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 $$W : [0, T] \times \Omega \to \mathbb{R}$$ 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


 * $$\operatorname{E} \left[ \left( \int_0^T X_t \, \mathrm{d} W_t \right)^2 \right] = \operatorname{E} \left[ \int_0^T X_t^2 \, \mathrm{d} t \right],$$

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

In other words, the Itô integral, as a function from the space $$L^2_{\mathrm{ad}} ([0,T] \times \Omega)$$ 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{align} ( X, Y )_{L^2_{\mathrm{ad}} ([0,T] \times \Omega)} & := \operatorname{E} \left( \int_0^T X_t \, Y_t \, \mathrm{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
 * $$\operatorname{E} \left[ \left( \int_0^T X_t \, \mathrm{d} W_t \right) \left( \int_0^T Y_t \, \mathrm{d} W_t \right) \right] = \operatorname{E} \left[ \int_0^T X_t Y_t \, \mathrm{d} t \right]$$

for $$X, Y \in L^2_{\mathrm{ad}} ([0,T] \times \Omega)$$.