Itō isometry

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

:$mathbb\{E\}\; left(\; int\_\{0\}^\{T\}\; X\_\{t\}\; ,\; mathrm\{d\}\; W\_\{t\}\; ight)^\{2\}\; =\; mathbb\{E\}\; left(\; int\_\{0\}^\{T\}\; X\_\{t\}^\{2\}\; ,\; mathrm\{d\}\; t\; ight),$

where $mathbb\{E\}$ denotes expectation with respect to classical Wiener measure $gamma$. In other words, the Itō stochastic integral, as a function

:Itō integrable processes $L^\{2\}\; (W)\; subset\; L^\{2\}\; (\; [0,\; T]\; imes\; Omega,\; mathcal\{B\}(\; [0,\; T]\; )\; otimes\; mathcal\{B\}(Omega),\; lambda\; otimes\; gamma;\; mathbb\{R\})\; o\; L^\{2\}\; (Omega,\; mathcal\{B\}(Omega),\; gamma;\; mathbb\{R\})$

is an isometry of normed vector spaces with respect to the norms induced by the inner products

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

and

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

References

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