Sigma-martingale

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. 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).

Mathematical definition
An $$\mathbb{R}^d$$-valued stochastic process $$X = (X_t)_{t = 0}^T$$ is a sigma-martingale if it is a semimartingale and there exists an $$\mathbb{R}^d$$-valued martingale M and an M-integrable predictable process $$\phi$$ with values in $$\mathbb{R}_+$$ such that
 * $$X = \phi \cdot M. $$