 Martingale representation theorem

A martingale representation theorem identifies driving martingales from which every martingale in a specified filtration can be reconstructed by stochastic integrals. The filtration and the class of integrable martingales are part of the assertion.

For the completed natural Brownian filtration, the Brownian martingale representation theorem gives $M_t=M_0+{\int }_0^t{H}_{s}d{W}_s$ for every square-integrable martingale, with a predictable integrand $H$ unique up to time-probability null sets. For example, $W_t^2-t={\int }_0^{t}2W_{s}d{W}_s$. In a filtration generated by independent Rademacher random variables, predictable representation in a Rademacher filtration gives the discrete analogue. Adding an independent source of randomness to a Brownian filtration may require another driving martingale; it does not preserve representation by the original Brownian motion alone.