Skorokhod integral

Summary

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:

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

Definition

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Preliminaries: the Malliavin derivative

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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.

The Skorokhod integral

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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

 

Properties

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  • 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  

Alternatives

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An alternative to the Skorokhod integral is the Ogawa integral.

References

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  1. ^ Hitsuda, Masuyuki (1972). "Formula for Brownian partial derivatives". Second Japan-USSR Symp. Probab. Th.2.: 111–114.
  2. ^ 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.