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

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

References

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