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| {{unreferenced|date=August 2009}}
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| {{Expert-subject|Mathematics|date=February 2009}}
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| In [[mathematics]], the '''Malliavin derivative''' is a notion of [[derivative]] in the [[Malliavin calculus]]. Intuitively, it is the notion of derivative appropriate to paths in [[classical Wiener space]], which are "usually" not differentiable in the usual sense. {{Citation Needed|date=August 2011}}
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| ==Definition==
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| Let <math>H</math> be the [[Cameron-Martin space]], and <math>C_{0}</math> denote [[classical Wiener space]]:
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| :<math>H := \{ f \in W^{1,2} ([0, T]; \mathbb{R}^{n}) \;|\; f(0) = 0 \} := \{ \text{paths starting at 0 with first derivative in } L^{2} \}</math>;
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| :<math>C_{0} := C_{0} ([0, T]; \mathbb{R}^{n}) := \{ \text{continuous paths starting at 0} \}</math>;
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| By the [[Sobolev_inequality#Sobolev_embedding_theorem|Sobolev embedding theorem]], <math>H \subset C_0</math>. Let
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| :<math>i : H \to C_{0}</math>
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| denote the [[inclusion map]].
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| Suppose that <math>F : C_{0} \to \mathbb{R}</math> is [[Fréchet derivative|Fréchet differentiable]]. Then the [[Fréchet derivative]] is a map
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| :<math>\mathrm{D} F : C_{0} \to \mathrm{Lin} (C_{0}; \mathbb{R})</math>;
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| i.e., for paths <math>\sigma \in C_{0}</math>, <math>\mathrm{D} F (\sigma)\;</math> is an element of <math>C_{0}^{*}</math>, the [[dual space]] to <math>C_{0}\;</math>. Denote by <math>\mathrm{D}_{H} F(\sigma)\;</math> the [[continuous function|continuous]] [[linear map]] <math>H \to \mathbb{R}</math> defined by
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| :<math>\mathrm{D}_{H} F (\sigma) := \mathrm{D} F (\sigma) \circ i : H \to \mathbb{R}, </math>
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| sometimes known as the [[H-derivative|''H''-derivative]]. Now define <math>\nabla_{H} F : C_{0} \to H</math> to be the [[adjoint]]{{dn|date=December 2013}} of <math>\mathrm{D}_{H} F\;</math> in the sense that
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| :<math>\int_0^T \left(\partial_t \nabla_H F(\sigma)\right) \cdot \partial_t h := \langle \nabla_{H} F (\sigma), h \rangle_{H} = \left( \mathrm{D}_{H} F \right) (\sigma) (h) = \lim_{t \to 0} \frac{F (\sigma + t i(h)) - F(\sigma)}{t}</math>.
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| Then the '''Malliavin derivative''' <math>\mathrm{D}_{t}</math> is defined by
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| :<math>\left( \mathrm{D}_{t} F \right) (\sigma) := \frac{\partial}{\partial t} \left( \left( \nabla_{H} F \right) (\sigma) \right).</math>
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| The [[domain (mathematics)|domain]] of <math>\mathrm{D}_{t}</math> is the set <math>\mathbf{F}</math> of all Fréchet differentiable real-valued functions on <math>C_{0}\;</math>; the [[codomain]] is <math>L^{2} ([0, T]; \mathbb{R}^{n})</math>.
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| The '''Skorokhod integral''' <math>\delta\;</math> is defined to be the [[adjoint]]{{dn|date=December 2013}} of the Malliavin derivative:
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| :<math>\delta := \left( \mathrm{D}_{t} \right)^{*} : \operatorname{image} \left( \mathrm{D}_{t} \right) \subseteq L^{2} ([0, T]; \mathbb{R}^{n}) \to \mathbf{F}^{*} = \mathrm{Lin} (\mathbf{F}; \mathbb{R}).</math>
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| ==See also== | |
| *[[H-derivative]]
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| ==References==
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| {{reflist}}
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| [[Category:Generalizations of the derivative]]
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| [[Category:Stochastic calculus]]
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Hi there, I am Felicidad Oquendo. Interviewing is what I do in my day occupation. Delaware is our birth place. The favorite pastime for him and his children is to drive and now he is trying to earn cash with it.
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