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.

Statement 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 $\alpha^{F} : [0, T] \times C_{0} \to \mathbb{R}$ (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 $B : [0, T] \times \Omega \to \mathbb{R}$ 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)