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)