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By construction, this implies that if is a martingale, then = will be an MDS—hence the name. The MDS is an extremely useful construct in modern probability theory because it implies much milder restrictions on the memory of the sequence than independence , yet most limit theorems that hold for an independent sequence will also hold for an MDS.
A convex function of a martingale is a submartingale, by Jensen's inequality. For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that X n 2 − n is a martingale). Similarly, a concave function of a martingale is a supermartingale.
A continuous semimartingale uniquely decomposes as X = M + A where M is a continuous local martingale and A is a continuous finite-variation process starting at zero. (Rogers & Williams 1987, p. 358) For example, if X is an Itō process satisfying the stochastic differential equation dX t = σ t dW t + b t dt, then
Differential signalling is a method for electrically transmitting information using two complementary signals. The technique sends the same electrical signal as a differential pair of signals, each in its own conductor. The pair of conductors can be wires in a twisted-pair or ribbon cable or traces on a printed circuit board.
Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential E ( X ) {\displaystyle {\mathcal {E}}(X)} of a continuous local martingale X {\displaystyle X} is a martingale are given by Kazamaki's condition , Novikov's condition , and Beneš's condition .
Uniform integrability is an extension to the notion of a family of functions being dominated in which is central in dominated convergence.Several textbooks on real analysis and measure theory use the following definition: [1] [2]
The condition that the martingale is bounded is essential; for example, an unbiased random walk is a martingale but does not converge. As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening.
The following result allows to express martingales as Itô integrals: if M is a square-integrable martingale on a time interval [0, T] with respect to the filtration generated by a Brownian motion B, then there is a unique adapted square integrable process on [0, T] such that = + almost surely, and for all t ∈ [0, T] (Rogers & Williams 2000 ...