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. 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).

Mathematical definition

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

$X=\varphi \cdot M.$

References

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