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] Consider a fixed probability space
and a Hilbert space
denotes expectation with respect to
Intuitively speaking, the Malliavin derivative of a random variable
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.
of the Hilbert space
is a Gaussian (normal) random variable, that the map taking
is a linear map, and that the mean and covariance structure is given by
, 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
, and then extending this definition to "smooth enough" random variables.
is smooth, the Malliavin derivative is defined using the earlier "formal definition" and the chain rule:
was a real-valued random variable, its derivative
-valued random variable, an element of the space
for "smooth" random variables, but an approximation procedure can be employed to define
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
consists of those processes
The Skorokhod integral of a process
is a real-valued random variable
was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for "simple processes": if
An alternative to the Skorokhod integral is the Ogawa integral.