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 Itō integral.
Statement of the theorem
Let (or simply for short) be classical Wiener space with classical Wiener measure . If , then there exists a unique Itō integrable process (i.e. in , where is canonical Brownian motion) such that
for -almost all .
In the above,
- is the expected value of ; and
- the integral is an Itō integral.
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 (i.e. a stochastic process whose law is Wiener measure). Let be the natural filtration of by the Brownian motion :
Suppose that is -measurable. Then there is a unique Itō integrable process such that
- -almost surely.
- Mao Xuerong. Stochastic differential equations and their applications. Chichester: Horwood. (1997)