Brownian martingale representation theorem

Let $({\mathcal{F}}_t)$ be the usual augmentation of the natural filtration of a standard Brownian motion $W$. On a finite horizon $T$, every $Z\in L^2({\mathcal{F}}_T)$ has the representationfor a predictable process $H$. Consequently every square-integrable martingale on $[0,T]$ has $M_t=M_0+{\int }_0^t{H}_{s}d{W}_s$. The Itô isometry makes $H$ unique up to $dtd\mathbb{P}$-almost everywhere equality. A proof first represents deterministic Gaussian exponentials, whose linear span is dense in L2 space, then uses the Itô isometry to close the class of represented terminal values.^[ref]

The natural-filtration qualification matters. In the filtration generated by independent Brownian motions $W,V$, both are martingales, but $V$ cannot be an Itô integral against $W$: its quadratic covariation with $W$ is zero, which would force the proposed integrand to vanish. A general Brownian filtration may therefore be too large.

In the completed natural Brownian filtration, a bounded continuously differentiable $\varphi$ with bounded derivative satisfiesThe integrand is unique up to $d\mathbb{P}dt$-almost everywhere equality. Gaussian integration by parts first gives the duality between the payoff and every square-integrable Itô integral. The Brownian martingale representation theorem then identifies the integrand as the projection of ${\varphi }^{\prime }(W_T)1_{\{t\le T\}}$ onto the predictable processes in the product $L^2$ space. This explains the conditional expectation in the formula, rather than a nonadapted terminal derivative.