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.^[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]

Mathematical definition

An [math]\displaystyle{ \mathbb{R}^d }[/math]-valued stochastic process [math]\displaystyle{ X = (X_t)_{t = 0}^T }[/math] is a sigma-martingale if it is a semimartingale and there exists an [math]\displaystyle{ \mathbb{R}^d }[/math]-valued martingale M and an M-integrable predictable process [math]\displaystyle{ \phi }[/math] with values in [math]\displaystyle{ \mathbb{R}_+ }[/math] such that

[math]\displaystyle{ X = \phi \cdot M. }[/math]^[1]

References

↑ ^1.0 ^1.1 F. Delbaen; W. Schachermayer (1998). "The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes". Mathematische Annalen 312 (2): 215–250. doi:10.1007/s002080050220. https://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0084.pdf. Retrieved October 14, 2011.
↑ Delbaen, Freddy; Schachermayer, Walter. "What is... a Free Lunch?". Notices of the AMS 51 (5): 526–528. http://www.ams.org/notices/200405/what-is.pdf. Retrieved October 14, 2011.

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