Sigma-martingale
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
of
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
, 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 of financial markets. In general, there exist two separate branches of finance that require ...
, sigma-martingales appear in the fundamental theorem of asset pricing as an equivalent condition to
no free lunch with vanishing risk No free lunch with vanishing risk (NFLVR) is a no-arbitrage argument. We have ''free lunch with vanishing risk'' if by utilizing a sequence of time self-financing portfolios, which converge to an arbitrage strategy, we can approximate a self-fina ...
(a no-
arbitrage In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between t ...
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. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
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 In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
predictable process In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits o ...
\phi with values in \mathbb_+ such that :X = \phi \cdot M.


References

{{probability-stub Martingale theory