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).

An ${\displaystyle \mathbb {R} ^{d}}$-valued stochastic process ${\displaystyle X:\Omega \times [0,T]\longrightarrow \mathbb {R} ^{d}}$ 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

where integration is understood in the sense of Ito calculus.