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 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[edit]

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[edit]

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