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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.
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.
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.
In the mathematical theory of probability, a Doob martingale (named after Joseph L. Doob, [1] also known as a Levy martingale) is a stochastic process that approximates a given random variable and has the martingale property with respect to the given filtration. It may be thought of as the evolving sequence of best approximations to the random ...
In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences. Suppose {: =,,,, …} is a martingale (or super-martingale) and
In mathematics, Doob's martingale inequality, also known as Kolmogorov’s submartingale inequality is a result in the study of stochastic processes.It gives a bound on the probability that a submartingale exceeds any given value over a given interval of time.
If is a continuous local martingale, then the Burkholder–Davis–Gundy inequality holds for any >. An alternative process, the predictable quadratic variation is sometimes used for locally square integrable martingales.
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.
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