In mathematics and information theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978.^{[1]} In financial mathematics, sigma-martingales appear in the fundamental theorem of asset pricing as an equivalent condition to no free lunch with vanishing risk (a no-arbitrage condition).^{[2]}
An -valued stochastic process is a sigma-martingale if it is a semimartingale and there exists an -valued martingale M and an M-integrable predictable process with values in such that