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 [math]\displaystyle{ W : [0, T] \times \Omega \to \mathbb{R} }[/math] denote the canonical real-valued Wiener process defined up to time [math]\displaystyle{ T \gt 0 }[/math], and let [math]\displaystyle{ X : [0, T] \times \Omega \to \mathbb{R} }[/math] be a stochastic process that is adapted to the natural filtration [math]\displaystyle{ \mathcal{F}_{*}^{W} }[/math] of the Wiener process.[clarification needed] Then
- [math]\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], }[/math]
where [math]\displaystyle{ \operatorname{E} }[/math] denotes expectation with respect to classical Wiener measure.
In other words, the Itô integral, as a function from the space [math]\displaystyle{ L^2_{\mathrm{ad}} ([0,T] \times \Omega) }[/math] of square-integrable adapted processes to the space [math]\displaystyle{ L^2 (\Omega) }[/math] of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products
- [math]\displaystyle{ \begin{align} ( X, Y )_{L^2_{\mathrm{ad}} ([0,T] \times \Omega)} & := \operatorname{E} \left( \int_0^T X_t \, Y_t \, \mathrm{d} t \right) \end{align} }[/math]
and
- [math]\displaystyle{ ( A, B )_{L^2 (\Omega)} := \operatorname{E} ( A B ) . }[/math]
As a consequence, the Itô integral respects these inner products as well, i.e. we can write
- [math]\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] }[/math]
for [math]\displaystyle{ X, Y \in L^2_{\mathrm{ad}} ([0,T] \times \Omega) }[/math] .
References
- Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1.
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