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.
In other words, the Itô integral, as a function from the space of square-integrable
adapted processes to the space of square-integrable random variables, is an
isometry of
normed vector spaces with respect to the norms induced by the
inner products
and
As a consequence, the Itô integral respects these inner products as well, i.e. we can write