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 Itō integral.

tatement of the theorem

Let $C\_\{0\}\; (\; [0,\; T]\; ;\; mathbb\{R\})$ (or simply $C\_\{0\}$ for short) be classical Wiener space with classical Wiener measure $gamma$. If $F\; in\; L^\{2\}\; (C\_\{0\};\; mathbb\{R\})$, then there exists a unique Itō integrable process $alpha^\{F\}\; :\; [0,\; T]\; imes\; C\_\{0\}\; o\; mathbb\{R\}$ (i.e. in $L^\{2\}\; (B)$, where $B$ is canonical Brownian motion) such that

:$F(sigma)\; =\; int\_\{C\_\{0\; F(p)\; ,\; mathrm\{d\}\; gamma\; (p)\; +\; int\_\{0\}^\{T\}\; alpha^\{F\}\; (sigma)\_\{t\}\; ,\; mathrm\{d\}\; sigma\_\{t\}$

for $gamma$-almost all $sigma\; in\; C\_\{0\}$.

In the above,
* $int\_\{C\_\{0\; F(p)\; ,\; mathrm\{d\}\; gamma\; (p)\; =\; mathbb\{E\}\; [F]$ is the expected value of $F$; and
* the integral $int\_\{0\}^\{T\}\; cdots,\; mathrm\{d\}\; sigma\_\{t\}$ 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 $(Omega,\; mathcal\{F\},\; mathbb\{P\})$ be a probability space. Let $B\; :\; [0,\; T]\; imes\; Omega\; o\; mathbb\{R\}$ be a Brownian motion (i.e. a stochastic process whose law is Wiener measure). Let $\{\; mathcal\{F\}\_\{t\}\; |\; 0\; leq\; t\; leq\; T\; \}$ be the natural filtration of $mathcal\{F\}$ by the Brownian motion $B$:::$mathcal\{F\}\_\{t\}\; =\; sigma\; \{\; B\_\{s\}^\{-1\}\; (A)\; |\; A\; in\; mathrm\{Borel\}\; (mathbb\{R\}),\; 0\; leq\; s\; leq\; t\; \}.$Suppose that $f\; in\; L^\{2\}\; (Omega;\; mathbb\{R\})$ is $mathcal\{F\}\_\{T\}$-measurable. Then there is a unique Itō integrable process $a^\{f\}\; in\; L^\{2\}\; (B)$ such that::$f\; =\; mathbb\{E\}\; [f]\; +\; int\_\{0\}^\{T\}\; a\_\{t\}^\{f\}\; ,\; mathrm\{d\}\; B\_\{t\}$ $mathbb\{P\}$-almost surely.

References

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