Base flow (random dynamical systems)

In mathematics, the Itō isometry, named after Kiyoshi Itō, is a crucial fact about Itō stochastic integrals. One of its main applications is to enable the computation of variances for stochastic processes.

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

$\mathbb{𝔼}\left[{\left(\underset{0}{\overset{T}{\int }}{X}_t\mathrm{d}{W}_t\right)}^2\right]=\mathbb{𝔼}\left[\underset{0}{\overset{T}{\int }}{X}_t^2\mathrm{d}t\right],$

where $\mathbb{𝔼}$ denotes expectation with respect to classical Wiener measure $\gamma$. In other words, the Itō stochastic integral, as a function, is an isometry of normed vector spaces with respect to the norms induced by the inner products

$(X,Y{)}_{L^2(W)}:=\mathbb{𝔼}\left(\underset{0}{\overset{T}{\int }}{X}_t\mathrm{d}{W}_t\underset{0}{\overset{T}{\int }}{Y}_t\mathrm{d}{W}_t\right)=\underset{\mathrm{\Omega }}{\int }\left(\underset{0}{\overset{T}{\int }}{X}_t\mathrm{d}{W}_t\underset{0}{\overset{T}{\int }}{Y}_t\mathrm{d}{W}_t\right)\mathrm{d}\gamma (\omega )$

and

$(A,B{)}_{L^2(\mathrm{\Omega })}:=\mathbb{𝔼}(AB)=\underset{\mathrm{\Omega }}{\int }A(\omega )B(\omega )\mathrm{d}\gamma (\omega ).$

References

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