Sigma-martingale

In probability theory, 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 $X:\mathrm{\Omega }\times [0,T]⟶{ℝ}^d$ 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

$X={\int }_0^{\cdot }\varphi dM,$

where integration is understood in the sense of Ito calculus.

References

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