Itō isometry

From Wikipedia, the free encyclopedia

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 $W:[0,T]\times \Omega \to {\mathbb {R}}$ 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 $\gamma$ . 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

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

and

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

References

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

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.