§ 08 / representation

Martingale Representation & Doob–Meyer

Representation of martingales, Lévy's characterization, and predictable quadratic variation.

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Section 1 / 7

The Martingale Representation Theorem

In Chapter 3 we showed that the Itô integral of an H2\Htwo-integrand is a continuous martingale. The Martingale Representation Theorem concerns the converse question:

For martingales with respect to the standard Brownian filtration that satisfy a square-integrability condition, the answer is yes.

Suppose that the representation (8.1) holds both with ϕ∈H2[0,T]\phi\in\Htwo[0,T] and with ψ∈H2[0,T]\psi\in\Htwo[0,T]. Then ϕ−ψ∈H2[0,T]\phi-\psi\in\Htwo[0,T] and

0=∫0T(ϕ(ω,s)−ψ(ω,s)) dBs.0 = \int_0^T (\phi(\omega,s)-\psi(\omega,s))\,dB_s.

Taking the second moment and applying the Itô isometry gives

0=E[∫0T(ϕ(ω,s)−ψ(ω,s))2 ds].0 = \E\left[ \int_0^T (\phi(\omega,s)-\psi(\omega,s))^2\,ds \right].

The integrand is nonnegative, so it must be zero P×dtP\times dt-almost everywhere. Therefore

ϕ=ψP×dt-a.e.\boxed{\phi=\psi\quad P\times dt\text{-a.e.}}

and uniqueness follows.

It is sufficient to prove the representation at time TT:

XT=x0+∫0Tϕ(ω,s) dBs.\boxed{ X_T=x_0+\int_0^T\phi(\omega,s)\,dB_s. }

Indeed, applying conditional expectation with respect to Ft\F_t gives

Xt=E[XT∣Ft]=x0+∫0tϕ(ω,s) dBs.X_t = \E[X_T\mid\F_t] = x_0+\int_0^t\phi(\omega,s)\,dB_s.

The proof proceeds in two steps:

  1. Prove (8.2) for a sufficiently large class of random variables.
  2. Show that this class is dense in
    L2(Ω,FT,P).L^2(\Omega,\F_T,P).

The result then follows by approximation.