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| In [[mathematics]], the '''Itō isometry''', named after [[Kiyoshi Itō]], is a crucial fact about [[Itō calculus|Itō stochastic integrals]]. One of its main applications is to enable the computation of [[variance]]s for [[stochastic processes]].
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| Let <math>W : [0, T] \times \Omega \to \mathbb{R}</math> denote the canonical real-valued [[Wiener process]] defined up to time <math>T > 0</math>, and let <math>X : [0, T] \times \Omega \to \mathbb{R}</math> be a stochastic process that is [[adapted process|adapted]] to the [[filtration (abstract algebra)|natural filtration]] <math>\mathcal{F}_{*}^{W}</math> of the Wiener process. Then
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| :<math>\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],</math>
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| where <math>\mathbb{E}</math> denotes [[expected value|expectation]] with respect to [[classical Wiener measure]] <math>\gamma</math>. In other words, the Itō stochastic integral, as a function, is an [[isometry]] of [[normed vector space]]s with respect to the norms induced by the [[inner product]]s
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| :<math>( 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)</math>
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| :<math>( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega).</math>
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| == References == | |
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| * {{cite book | author=Øksendal, Bernt K. | authorlink=Bernt Øksendal | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer, Berlin | year=2003 | isbn=3-540-04758-1}}
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| {{DEFAULTSORT:Ito Isometry}}
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| [[Category:Stochastic calculus]]
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