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 ${\displaystyle W:[0,T]\times \Omega \to \mathbb {R} }$ denote the canonical real-valued Wiener process defined up to time ${\displaystyle T>0}$, and let ${\displaystyle X:[0,T]\times \Omega \to \mathbb {R} }$ be a stochastic process that is adapted to the natural filtration ${\displaystyle {\mathcal {F}}_{*}^{W}}$ of the Wiener process. Then

${\displaystyle \operatorname {E} \left[\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\right)^{2}\right]=\operatorname {E} \left[\int _{0}^{T}X_{t}^{2}\,\mathrm {d} t\right],}$

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

In other words, the Itô integral, as a function from the space ${\displaystyle L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}$ of square-integrable adapted processes to the space ${\displaystyle L^{2}(\Omega )}$ of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products

${\displaystyle {\begin{aligned}(X,Y)_{L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}&:=\operatorname {E} \left(\int _{0}^{T}X_{t}\,Y_{t}\,\mathrm {d} t\right)\end{aligned}}}$

and

${\displaystyle (A,B)_{L^{2}(\Omega )}:=\operatorname {E} (AB).}$

As a consequence, the Itô integral respects these inner products as well, i.e. we can write

${\displaystyle \operatorname {E} \left[\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\right)\left(\int _{0}^{T}Y_{t}\,\mathrm {d} W_{t}\right)\right]=\operatorname {E} \left[\int _{0}^{T}X_{t}Y_{t}\,\mathrm {d} t\right]}$

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