Sigma-martingale
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
information theory Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, ...
of
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
, a sigma-martingale is a
semimartingale In probability theory, a real-valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. In
financial mathematics Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the Finance#Quantitative_finance, financial field. In general, there exist two separate ...
, sigma-martingales appear in the
fundamental theorem of asset pricing The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An a ...
as an equivalent condition to
no free lunch with vanishing risk No free lunch with vanishing risk (NFLVR) is a concept used in mathematical finance as a strengthening of the no-arbitrage condition. In continuous time finance the existence of an equivalent martingale measure (EMM) is no more equivalent to the ...
(a no-
arbitrage Arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more marketsstriking a combination of matching deals to capitalize on the difference, the profit being the difference between the market prices at which th ...
condition).


Mathematical definition

An \mathbb^d-valued
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
X = (X_t)_^T is a ''sigma-martingale'' if it is a
semimartingale In probability theory, a real-valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
and there exists an \mathbb^d-valued martingale ''M'' and an ''M''- integrable predictable process \phi with values in \mathbb_+ such that :X = \phi \cdot M.


References

Martingale theory {{probability-stub