Integral representation theorem for classical Wiener space


In mathematics, the integral representation theorem for classical Wiener space is a result in the fields of measure theory and stochastic analysis. Essentially, it shows how to decompose a function on classical Wiener space into the sum of its expected value and an Itô integral.

Statement of the theorem

Let be classical Wiener space with classical Wiener measure. If, then there exists a unique Itô integrable process such that
for -almost all.
In the above,
The proof of the integral representation theorem requires the Clark-Ocone theorem from the Malliavin calculus.

Corollary: integral representation for an arbitrary probability space

Let be a probability space. Let be a Brownian motion. Let be the natural filtration of by the Brownian motion :
Suppose that is -measurable. Then there is a unique Itô integrable process such that