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 $\mathbb {R} ^{d))$-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

- $X=\phi \cdot M.$
^{[1]}

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^{a}^{b}F. Delbaen; W. Schachermayer (1998). "The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes" (PDF).*Mathematische Annalen*.**312**(2): 215–250. doi:10.1007/s002080050220. S2CID 18366067. Retrieved October 14, 2011. **^**Delbaen, Freddy; Schachermayer, Walter. "What is... a Free Lunch?" (PDF).*Notices of the AMS*.**51**(5): 526–528. Retrieved October 14, 2011.

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