Martingles

Martingles

Martingales. Definitions and properties The theory of martingales plays a very important ans ueful role in the study of stochastic processes. A formal definition is given below. Definition Let (Ω,F,P) be a probability space. A martingale se-quence of length nis a chain X 1,X 2,···,X n of random variables and corre-sponding sub Examples of martingales. Let X t+1 = X t ± b t where +b t and -b t occur with equal probability b t is measurable ℱ t, and the outcome ±b t is measurable ℱ t+1 (in other words, my "bet" b t can only depend on what has happened so far and not on the future, but my knowledge ℱ t includes the outcome of all past bets) Lecture 3: Martingales: definition, examples 2 2 Examples EX (Sums of iid RVs with mean 0) Let X 0;X 1;iid RVs integrable and centered with X 0 = 0 F n = ˙(X 0;;X n) S n = P i n X i Then note that EjS nj

Martingale - Wikipedia

com The European Mathematical Society, martingles. with probability 1. Related notions are stochastic processes which form a submartingale, if. Example 1. Example 2. In the gambling world such a system is called a martingale, which explains the origin of the mathematical term "martingale".

As a particular case of this the Wald identity follows:. are martingales. which is a martingles martingale. In the case of continuous time the Doob, martingles, Burkholder and Davis martingles are still true for right-continuous processes having left limits.

Stopping times are also called optimal times, or, in the older literature, Markov martingles or Markov moments, cf. Markov moment, martingles.

The optimal sampling theorem is also called the stopping theorem or Doob's stopping theorem. The notion of a martingale is one of the most important concepts in modern probability theory.

It is basic in the theories martingles Markov processes and stochastic integrals, and is useful in many parts of analysis convergence theorems in ergodic theory, derivatives and lifting in measure theory, inequalities in the theory of singular integrals, etc. Log in, martingles. Navigation Main page Pages A-Z StatProb Collection Recent changes Current events Random page Help Project talk Request account. Tools What links here Related changes Special pages Printable version Permanent link Page information.

Namespaces Page Discussion. Views View View source History. Jump to: navigation martingles, search. References [D] J.

Martingles, A. Skorohod, "The theory of stochastic processes"martingles, 1Springer Translated from Russian MR Zbl References [DM] C. Dellacherie, P.

Meyer, "Probabilities and potential"1—3North-Holland — pp. Theory of martingales Translated from French MR MR MR MR MR MR Zbl Doob, "Classical potential theory and its probabilistic counterpart"martingles, Springer pp. Neveu, "Discrete-parameter martingales"North-Holland Translated from French MR Zbl Ville, martingles, "Etude critique de la notion de collectif"Gauthier-Villars Zbl Wall, C.

Heyde, "Martingale limit theory and its application"Acad. Press MR Martingles to Cite This Entry: Martingale. Encyclopedia of Mathematics. This article was adapted from an original article by A, martingles. Shiryaev originatorwhich appeared in Encyclopedia of Mathematics - ISBN See original article. Categories : TeX auto TeX done Probability and statistics Probability theory and stochastic processes Stochastic processes, martingles.

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106 (a) - Martingales

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Martingale System

Examples of martingales. Let X t+1 = X t ± b t where +b t and -b t occur with equal probability b t is measurable ℱ t, and the outcome ±b t is measurable ℱ t+1 (in other words, my "bet" b t can only depend on what has happened so far and not on the future, but my knowledge ℱ t includes the outcome of all past bets) Lecture 3: Martingales: definition, examples 2 2 Examples EX (Sums of iid RVs with mean 0) Let X 0;X 1;iid RVs integrable and centered with X 0 = 0 F n = ˙(X 0;;X n) S n = P i n X i Then note that EjS nj Martingale (collar) for dogs and other animals. Martingale (betting system), in 18th century France. a dolphin striker, a spar aboard a sailing ship. In the sport of fencing, a martingale is a strap attached to the sword handle to prevent a sword from being dropped if disarmed. In the theatrical lighting industry, martingale is an obsolete term Estimated Reading Time: 40 secs