Theorem (Itô Representation theorem)

Let $(B_{t})_{t\ge 0}$ be standard B.M., let $(\mathcal{F}_{t})_{t\ge 0}$ be the completed Natural Filtration of $(B_{t})_{t\ge 0}$ and let $\mathcal{F}_{\infty}=\sigma(\bigcup_{t\in\mathbb{R}^{+}}\mathcal{F}_{t})$. Let $F\in L^{2}(\Omega,\mathcal{F}_{\infty},P)$ (where $\mathcal{F}_{\infty}\subset \mathcal{F}$). Then $$\exists!H\in L^{2}(\mathbb{R}^{+}\times\Omega,\mathscr{P},\mu_{B}):F=E[F]+\int\limits H \, dB$$