Brownian martingale representation theorem

Every square-integrable random variable measurable with respect to a Brownian filtration can be written as its expectation plus an Itô integral against that Brownian motion. Equivalently, every square-integrable martingale in that filtration has the formfor a predictable square-integrable process $H$.

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.