In the mathematical theory of probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, sigma-martingale is a semimartingale

Semimartingale

In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

 with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. It should be noted that not every local martingale

Local martingale

In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; however, in general a local martingale is not a martingale, because its...

 is a sigma-martingale. 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

Arbitrage

In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...

 condition).

Mathematical definition

An -valued stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

  is a sigma-martingale if it is a semimartingale

Semimartingale

In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

 and there exists an -valued martingale

Martingale

Martingale can refer to:*Martingale , a stochastic process in which the conditional expectation of the next value, given the current and preceding values, is the current value*Martingale for horses...

 M and an M-integrable predictable process

Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process which the value is knowable at a prior time...

with values in such that