Itô isometry について

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Itō isometry ：ウィキペディア英語版

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 stochastic processes.
Let

) \times \Omega \to \mathbb

denote the canonical real-valued Wiener process defined up to time

T > 0

, and let

) \times \Omega \to \mathbb

be a stochastic process that is adapted to the natural filtration

\mathcal_^

of the Wiener process. Then
:

\mathbb \left(\left( \int_0^T X_t \, \mathrm W_t \right)^2 \right) = \mathbb \left(\int_0^T X_t^2 \, \mathrm t \right),

where

\mathbb

\gamma

. In other words, the Itô stochastic integral, as a function, is an isometry of normed vector spaces with respect to the norms induced by the inner products
:

)& = \int_\Omega \left( \int_0^T X_t \, \mathrm W_t \int_0^T Y_t \, \mathrm W_t \right) \, \mathrm \gamma (\omega)\end

and
:

( A, B )_ := \mathbb ( A B ) = \int_\Omega A(\omega) B(\omega) \, \mathrm \gamma (\omega).

== References ==

*

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
■ウィキペディアで「Itô isometry」の詳細全文を読む

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