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 {\displaystyle \mathbb {R} ^{d}} -valued stochastic process X : Ω × [ 0 , T ] ⟶ R d {\displaystyle X:\Omega \times [0,T]\longrightarrow \mathbb {R} ^{d}} is a sigma-martingale if it is a semimartingale and there exists an R d {\displaystyle \mathbb {R} ^{d}} -valued martingale M and an M-integrable predictable process ϕ {\displaystyle \phi } with values in R + {\displaystyle \mathbb {R} _{+}} such that
X = ∫ 0 ⋅ ϕ d M , {\displaystyle X=\int _{0}^{\cdot }\phi dM,}
where integration is understood in the sense of Ito calculus.
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