Integral representation theorem for classical Wiener space

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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[edit]

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[edit]

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.

References[edit]

  • Mao Xuerong. Stochastic differential equations and their applications. Chichester: Horwood. (1997)