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| In [[mathematics]], the '''Skorokhod integral''', often denoted ''δ'', is an [[Operator (mathematics)|operator]] of great importance in the theory of [[stochastic processes]]. It is named after the [[Ukraine|Ukrainian]] [[mathematician]] [[Anatoliy Skorokhod]]. Part of its importance is that it unifies several concepts:
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| * ''δ'' is an extension of the [[Itō integral]] to non-[[adapted process]]es;
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| * ''δ'' is the [[adjoint operator|adjoint]] of the [[Malliavin derivative]], which is fundamental to the stochastic [[calculus of variations]] ([[Malliavin calculus]]);
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| * ''δ'' is an infinite-dimensional generalization of the [[divergence]] operator from classical [[vector calculus]].
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| ==Definition==
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| ===Preliminaries: the Malliavin derivative===
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| Consider a fixed [[probability space]] (Ω, Σ, '''P''') and a [[Hilbert space]] ''H''; '''E''' denotes [[expected value|expectation]] with respect to '''P'''
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| :<math>\mathbf{E} [X] := \int_{\Omega} X(\omega) \, \mathrm{d} \mathbf{P}(\omega).</math>
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| Intuitively speaking, the Malliavin derivative of a random variable ''F'' in ''L''<sup>''p''</sup>(Ω) is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of ''H'' and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin derivative.
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| Consider a family of '''R'''-valued [[random variables]] ''W''(''h''), indexed by the elements ''h'' of the Hilbert space ''H''. Assume further that each ''W''(''h'') is a Gaussian ([[normal distribution|normal]]) random variable, that the map taking ''h'' to ''W''(''h'') is a [[linear map]], and that the [[expected value|mean]] and [[covariance]] structure is given by
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| :<math>\mathbf{E} [W(h)] = 0,</math> | |
| :<math>\mathbf{E} [W(g)W(h)] = \langle g, h \rangle_{H},</math>
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| for all ''g'' and ''h'' in ''H''. It can be shown that, given ''H'', there always exists a probability space (Ω, Σ, '''P''') and a family of random variables with the above properties. The Malliavin derivative is essentially defined by formally setting the derivative of the random variable ''W''(''h'') to be ''h'', and then extending this definition to “[[smooth function|smooth enough]]” random variables. For a random variable ''F'' of the form
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| :<math>F = f(W(h_{1}), \ldots, W(h_{n})),</math>
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| where ''f'' : '''R'''<sup>''n''</sup> → '''R''' is smooth, the '''Malliavin derivative''' is defined using the earlier “formal definition” and the chain rule:
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| :<math>\mathrm{D} F := \sum_{i = 1}^{n} \frac{\partial f}{\partial x_{i}} (W(h_{1}), \ldots, W(h_{n})) h_{i}.</math>
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| In other words, whereas ''F'' was a real-valued random variable, its derivative D''F'' is an ''H''-valued random variable, an element of the space ''L''<sup>''p''</sup>(Ω;''H''). Of course, this procedure only defines D''F'' for “smooth” random variables, but an approximation procedure can be employed to define D''F'' for ''F'' in a large subspace of ''L''<sup>''p''</sup>(Ω); the [[domain (mathematics)|domain]] of D is the [[closure (topology)|closure]] of the smooth random variables in the [[seminorm]] :
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| <math>\| F \|_{1, p} := \big( \mathbf{E}[|F|^{p}] + \mathbf{E}[\| \mathrm{D}F \|_{H}^{p}] \big)^{1/p}.</math>
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| This space is denoted by '''D'''<sup>1,''p''</sup> and is called the [[Watanabe-Sobolev space]].
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| ===The Skorokhod integral===
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| For simplicity, consider now just the case ''p'' = 2. The '''Skorokhod integral''' ''δ'' is defined to be the ''L''<sup>2</sup>-adjoint of the Malliavin derivative D. Just as D was not defined on the whole of ''L''<sup>2</sup>(Ω), ''δ'' is not defined on the whole of ''L''<sup>2</sup>(Ω; ''H''): the domain of ''δ'' consists of those processes ''u'' in ''L''<sup>2</sup>(Ω; ''H'') for which there exists a constant ''C''(''u'') such that, for all ''F'' in '''D'''<sup>1,2</sup>,
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| :<math>\big| \mathbf{E} [ \langle \mathrm{D} F, u \rangle_{H} ] \big| \leq C(u) \| F \|_{L^{2} (\Omega)}.</math>
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| The '''Skorokhod integral''' of a process ''u'' in ''L''<sup>2</sup>(Ω; ''H'') is a real-valued random variable ''δu'' in ''L''<sup>2</sup>(Ω); if ''u'' lies in the domain of ''δ'', then ''δu'' is defined by the relation that, for all ''F'' ∈ '''D'''<sup>1,2</sup>,
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| :<math>\mathbf{E} [F \, \delta u] = \mathbf{E} [ \langle \mathrm{D}F, u \rangle_{H} ].</math>
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| Just as the Malliavin derivative D was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for “simple processes”: if ''u'' is given by
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| :<math>u = \sum_{j = 1}^{n} F_{j} h_{j}</math>
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| with ''F''<sub>''j''</sub> smooth and ''h''<sub>''j''</sub> in ''H'', then | |
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| :<math>\delta u = \sum_{j = 1}^{n} \left( F_{j} W(h_{j}) - \langle \mathrm{D} F_{j}, h_{j} \rangle_{H} \right).</math>
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| ==Properties==
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| * The [[isometry]] property: for any process ''u'' in ''L''<sup>2</sup>(Ω; ''H'') that lies in the domain of ''δ'',
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| ::<math>\mathbf{E} \big[ (\delta u)^{2} \big] = \mathbf{E} \big[ \| u \|_{H}^{2} \big] + \mathbf{E} \big[ \| \mathrm{D} u \|_{H \otimes H}^{2} \big].</math>
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| :If ''u'' is an adapted process, then the second term on the right-hand side is zero, the Skorokhod and Itō integrals coincide, and the above equation becomes the [[Itō isometry]].
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| * The derivative of a Skorokhod integral is given by the formula
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| ::<math>\mathrm{D}_{h} (\delta u) = \langle u, h \rangle_{H} + \delta (\mathrm{D}_{h} u),</math>
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| :where D<sub>''h''</sub>''X'' stands for (D''X'')(''h''), the random variable that is the value of the process D''X'' at “time” ''h'' in ''H''.
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| * The Skorokhod integral of the product of a random variable ''F'' in '''D'''<sup>1,2</sup> and a process ''u'' in dom(''δ'') is given by the formula
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| ::<math>\delta (F u) = F \, \delta u - \langle \mathrm{D} F, u \rangle_{H}.</math>
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| ==References==
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| * {{springer|title=Skorokhod integral|id=p/s110170}}
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| * {{cite book
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| | last = Ocone
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| | first = Daniel L.
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| | chapter = A guide to the stochastic calculus of variations
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| | title = Stochastic analysis and related topics (Silivri, 1986)
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| | series = Lecture Notes in Math. 1316
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| | pages = 1–79
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| | publisher = Springer
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| | location = Berlin
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| | year = 1988
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| }} {{MathSciNet|id=953793}}
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| * {{cite web
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| | last = Sanz-Solé
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| | first = Marta
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| | title = Applications of Malliavin Calculus to Stochastic Partial Differential Equations (Lectures given at Imperial College London, 7–11 July 2008)
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| | year = 2008
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| | url = http://www.ma.ic.ac.uk/~dcrisan/lecturenotes-london.pdf
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| | accessdate = 2008-07-09
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| }}
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| [[Category:Definitions of mathematical integration]]
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| [[Category:Stochastic calculus]]
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