Itō isometry
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In mathematics, the Itō isometry 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 \Omega \to \mathbb{R}](../../../math/8/4/c/84c8a10f505aa54305540b6f4e32f160.png)
denote the canonical real-valued Wiener process defined up to time T > 0, and let ![X : [0, T] \times \Omega \to \mathbb{R}](../../../math/0/7/e/07e47195106bc673d3616c50a77cdc8f.png)
be a stochastic process that is adapted to the natural filtration 
of the Wiener process. Then
where 
denotes expectation with respect to classical Wiener measure γ. In other words, the Itō stochastic integral, as a function
- Itō integrable processes
![L^{2} (W) \subset L^{2} ([0, T] \times \Omega, \mathcal{B}([0, T]) \otimes \mathcal{B}(\Omega), \lambda \otimes \gamma; \mathbb{R}) \to L^{2} (\Omega, \mathcal{B}(\Omega), \gamma; \mathbb{R})](../../../math/e/6/4/e64c12b756ffb9b5971825172a846478.png)
is an isometry of normed vector spaces with respect to the norms induced by the inner products
and
[edit] References
- Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1.
