Preliminaries: the Malliavin derivative
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
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