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 .
