Sigma-martingale - Wikipedia

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

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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]

References

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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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