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 Itô integral.

Statement of the theorem

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

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

for ${\displaystyle \gamma }$-almost all ${\displaystyle \sigma \in C_{0}}$.

In the above,

• ${\displaystyle \int _{C_{0}}F(p)\,\mathrm {d} \gamma (p)=\mathbb {E} [F]}$ is the expected value of ${\displaystyle F}$; and
• the integral ${\displaystyle \int _{0}^{T}\cdots \,\mathrm {d} \sigma _{t}}$ is an Itô 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 ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ be a probability space. Let ${\displaystyle B:[0,T]\times \Omega \to \mathbb {R} }$ be a Brownian motion (i.e. a stochastic process whose law is Wiener measure). Let ${\displaystyle \{{\mathcal {F}}_{t}|0\leq t\leq T\}}$ be the natural filtration of ${\displaystyle {\mathcal {F}}}$ by the Brownian motion ${\displaystyle B}$:

${\displaystyle {\mathcal {F}}_{t}=\sigma \{B_{s}^{-1}(A)|A\in \mathrm {Borel} (\mathbb {R} ),0\leq s\leq t\}.}$

Suppose that ${\displaystyle f\in L^{2}(\Omega ;\mathbb {R} )}$ is ${\displaystyle {\mathcal {F}}_{T}}$-measurable. Then there is a unique Itô integrable process ${\displaystyle a^{f}\in L^{2}(B)}$ such that

${\displaystyle f=\mathbb {E} [f]+\int _{0}^{T}a_{t}^{f}\,\mathrm {d} B_{t}}$ ${\displaystyle \mathbb {P} }$-almost surely.

References

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