A martingale representation theorem states conditions under which every martingale in a filtration can be represented as a stochastic integral with respect to one or more fundamental martingales.
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 .
In the completed natural Brownian filtration, a bounded continuously differentiable with bounded derivative satisfiesThe integrand is unique up to -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 onto the predictable processes in the product space. This explains the conditional expectation in the formula, rather than a nonadapted terminal derivative.
