Martingale Representation Theorem
1 Martingale Representation Theorem
The martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.
Let \(B(t)\) be \(n\)-dimensional BM. Suppose \(M_{t}\) is an \(\mathcal{F}_{t}^{(n)}\)-martingale (w.r.t.) \(\mathbb{P}\) and that \(M_{t}\in\mathcal{L}_{2}(\mathbb{P})\) for all \(t \geq 0\). Then there exists a unique stochastic process \(g(s,\omega)\) such that \(g \in \mathcal{V}^{(n)}(0,t)\) for all \(t \geq 0\) and \[ M_{t}(\omega)=\mathbb{E}[M_{0}]+\int_{0}^t g(s,\omega) dB(s), \] almost surely for all \(g \geq 0\).
The proof of this result is involved and is omitted from this note.