World Library  
Flag as Inappropriate
Email this Article

Skorokhod integral

Article Id: WHEBN0018420736
Reproduction Date:

Title: Skorokhod integral  
Author: World Heritage Encyclopedia
Language: English
Subject: Malliavin calculus, Stochastic calculus, Paley–Wiener integral, Clark–Ocone theorem, Burkill integral
Collection: Definitions of Mathematical Integration, Stochastic Calculus
Publisher: World Heritage Encyclopedia

Skorokhod integral

In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod. Part of its importance is that it unifies several concepts:


  • Definition 1
    • Preliminaries: the Malliavin derivative 1.1
    • The Skorokhod integral 1.2
  • Properties 2
  • References 3


Preliminaries: the Malliavin derivative

Consider a fixed probability space (Ω, Σ, P) and a Hilbert space H; E denotes expectation with respect to P

\mathbf{E} [X] := \int_{\Omega} X(\omega) \, \mathrm{d} \mathbf{P}(\omega).

Intuitively speaking, the Malliavin derivative of a random variable F in Lp(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.

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) random variable, that the map taking h to W(h) is a linear map, and that the mean and covariance structure is given by

\mathbf{E} [W(h)] = 0,
\mathbf{E} [W(g)W(h)] = \langle g, h \rangle_{H},

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 enough” random variables. For a random variable F of the form

F = f(W(h_{1}), \ldots, W(h_{n})),

where f : Rn → R is smooth, the Malliavin derivative is defined using the earlier “formal definition” and the chain rule:

\mathrm{D} F := \sum_{i = 1}^{n} \frac{\partial f}{\partial x_{i}} (W(h_{1}), \ldots, W(h_{n})) h_{i}.

In other words, whereas F was a real-valued random variable, its derivative DF is an H-valued random variable, an element of the space Lp(Ω;H). Of course, this procedure only defines DF for “smooth” random variables, but an approximation procedure can be employed to define DF for F in a large subspace of Lp(Ω); the domain of D is the closure of the smooth random variables in the seminorm :

\| F \|_{1, p} := \big( \mathbf{E}[|F|^{p}] + \mathbf{E}[\| \mathrm{D}F \|_{H}^{p}] \big)^{1/p}.

This space is denoted by D1,p and is called the Watanabe-Sobolev space.

The Skorokhod integral

For simplicity, consider now just the case p = 2. The Skorokhod integral δ is defined to be the L2-adjoint of the Malliavin derivative D. Just as D was not defined on the whole of L2(Ω), δ is not defined on the whole of L2(Ω; H): the domain of δ consists of those processes u in L2(Ω; H) for which there exists a constant C(u) such that, for all F in D1,2,

\big| \mathbf{E} [ \langle \mathrm{D} F, u \rangle_{H} ] \big| \leq C(u) \| F \|_{L^{2} (\Omega)}.

The Skorokhod integral of a process u in L2(Ω; H) is a real-valued random variable δu in L2(Ω); if u lies in the domain of δ, then δu is defined by the relation that, for all F ∈ D1,2,

\mathbf{E} [F \, \delta u] = \mathbf{E} [ \langle \mathrm{D}F, u \rangle_{H} ].

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

u = \sum_{j = 1}^{n} F_{j} h_{j}

with Fj smooth and hj in H, then

\delta u = \sum_{j = 1}^{n} \left( F_{j} W(h_{j}) - \langle \mathrm{D} F_{j}, h_{j} \rangle_{H} \right).


  • The isometry property: for any process u in L2(Ω; H) that lies in the domain of δ,
\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].
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.
  • The derivative of a Skorokhod integral is given by the formula
\mathrm{D}_{h} (\delta u) = \langle u, h \rangle_{H} + \delta (\mathrm{D}_{h} u),
where DhX stands for (DX)(h), the random variable that is the value of the process DX at “time” h in H.
  • The Skorokhod integral of the product of a random variable F in D1,2 and a process u in dom(δ) is given by the formula
\delta (F u) = F \, \delta u - \langle \mathrm{D} F, u \rangle_{H}.


  • Hazewinkel, Michiel, ed. (2001), "Skorokhod integral",  
  • Ocone, Daniel L. (1988). "A guide to the stochastic calculus of variations". Stochastic analysis and related topics (Silivri, 1986). Lecture Notes in Math. 1316. Berlin: Springer. pp. 1–79.  MR 953793
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.