Itô isometry

Article summary

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.…

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