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