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

- is an extension of the Itô integral to non-adapted processes;
- is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus);
- is an infinite-dimensional generalization of the divergence operator from classical vector calculus.

The integral was introduced by Hitsuda in 1972^{[1]} and by Skorokhod in 1975.^{[2]}

Consider a fixed probability space and a Hilbert space ; denotes expectation with respect to

Intuitively speaking, the Malliavin derivative of a random variable in is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin derivative.

Consider a family of -valued random variables , indexed by the elements of the Hilbert space . Assume further that each is a Gaussian (normal) random variable, that the map taking to is a linear map, and that the mean and covariance structure is given by

for all and in . It can be shown that, given , there always exists a probability space 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 to be , and then extending this definition to "smooth enough" random variables. For a random variable of the form

where is smooth, the **Malliavin derivative** is defined using the earlier "formal definition" and the chain rule:

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

This space is denoted by and is called the Watanabe–Sobolev space.

For simplicity, consider now just the case . The **Skorokhod integral** is defined to be the -adjoint of the Malliavin derivative . Just as was not defined on the whole of , is not defined on the whole of : the domain of consists of those processes in for which there exists a constant such that, for all in ,

The **Skorokhod integral** of a process in is a real-valued random variable in ; if lies in the domain of , then is defined by the relation that, for all ,

Just as the Malliavin derivative was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for "simple processes": if is given by

with smooth and in , then

- The isometry property: for any process in that lies in the domain of , If is an adapted process, then for , so the second term on the right-hand side vanishes. The Skorokhod and Itô integrals coincide in that case, and the above equation becomes the Itô isometry.
- The derivative of a Skorokhod integral is given by the formula where stands for , the random variable that is the value of the process at "time" in .
- The Skorokhod integral of the product of a random variable in and a process in is given by the formula

An alternative to the Skorokhod integral is the Ogawa integral.

**^**Hitsuda, Masuyuki (1972). "Formula for Brownian partial derivatives".*Second Japan-USSR Symp. Probab. Th.2.*: 111–114.**^**Kuo, Hui-Hsiung (2014). "The Itô calculus and white noise theory: a brief survey toward general stochastic integration".*Communications on Stochastic Analysis*.**8**(1). doi:10.31390/cosa.8.1.07.

- "Skorokhod integral",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - 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. MR953793 - Sanz-Solé, Marta (2008). "Applications of Malliavin Calculus to Stochastic Partial Differential Equations (Lectures given at Imperial College London, 7–11 July 2008)" (PDF). Retrieved 2008-07-09.