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 Itô integral.

Statement of the theorem

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

[math]\displaystyle{ F(\sigma) = \int_{C_{0}} F(p) \, \mathrm{d} \gamma (p) + \int_{0}^{T} \alpha^{F} (\sigma)_{t} \, \mathrm{d} \sigma_{t} }[/math]

for [math]\displaystyle{ \gamma }[/math]-almost all [math]\displaystyle{ \sigma \in C_{0} }[/math].

In the above,

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

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

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

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

References

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




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