Editing Malliavin calculus (section)

== Skorokhod integral ==
{{Main|Skorokhod integral}}
The [[Skorokhod integral]] operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative in the [[white noise]] case when the Hilbert space is an <math>L^2</math> space, thus for u in the domain of the operator which is a subset of <math>L^2([0,\infty) \times \Omega)</math>,
for '''F''' in the domain of the Malliavin derivative, we require

: <math> E (\langle DF, u \rangle ) = E (F \delta (u) ),</math>

where the inner product is that on <math>L^2[0,\infty)</math> viz

: <math> \langle f, g \rangle = \int_0^\infty f(s) g(s) \, ds.</math>

The existence of this adjoint follows from the [[Riesz representation theorem]] for linear operators on [[Hilbert spaces]].

It can be shown that if ''u'' is adapted then

: <math> \delta(u) = \int_0^\infty u_t\, d W_t ,</math>

where the integral is to be understood in the Itô sense.   Thus this provides a method of extending the Itô integral to non adapted integrands.