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

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

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

In other words, the Itô integral, as a function from the space $L_{\mathrm {ad} }^{2}([0,T]\times \Omega )$ 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{aligned}(X,Y)_{L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}&:=\mathbb {E} \left(\int _{0}^{T}X_{t}\,Y_{t}\,\mathrm {d} t\right)\end{aligned}}

and

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

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

\mathbb {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]=\mathbb {E} \left[\int _{0}^{T}X_{t}Y_{t}\,\mathrm {d} t\right]

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

References

Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1.

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