Sigma-martingale explained

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).^[1]

Mathematical definition

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 ^[2]

Notes and References

1. What is... a Free Lunch?. Freddy. Delbaen. Walter. Schachermayer. Notices of the AMS. 51. 5. 526–528. October 14, 2011.
2. The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes. F. Delbaen. W. Schachermayer. Mathematische Annalen. 1998. 312. 2 . 215–250. October 14, 2011. 10.1007/s002080050220. 18366067 .