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 ] × Ω → R denote the canonical real-valued Wiener process defined up to time T > 0 , and let X : [ 0 , T ] × Ω → R be a stochastic process that is adapted to the natural filtration F ∗ W of the Wiener process. Then
E [ ( ∫ 0 T X t d W t ) 2 ] = E [ ∫ 0 T X t 2 d t ] , where E denotes expectation with respect to classical Wiener measure.
In other words, the Itô integral, as a function from the space L a d 2 ( [ 0 , T ] × Ω ) of square-integrable adapted processes to the space L 2 ( Ω ) of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products
( X , Y ) L a d 2 ( [ 0 , T ] × Ω ) := E ( ∫ 0 T X t Y t d t ) and
( A , B ) L 2 ( Ω ) := E ( A B ) . As a consequence, the Itô integral respects these inner products as well, i.e. we can write
E [ ( ∫ 0 T X t d W t ) ( ∫ 0 T Y t d W t ) ] = E [ ∫ 0 T X t Y t d t ] for X , Y ∈ L a d 2 ( [ 0 , T ] × Ω ) .
