Martingale (probability theory)

In probability theory a martingale is a stochastic process in which the conditional expected value of an observation is equal to the value of the previous observation.

Martingales were introduced by Paul Lévy in mathematics.

Definition

A time-discrete case

On a probability space is a sequence of integrable random variables given, that is, applicable to all. This sequence is called a martingale if for all the conditional expectation of a future observation is equal to the last observed value, ie

This condition can be interpreted such that a martingale is a fair game, since the expected value of a future observation is equal to the last observation made ​​. If the value of the time martingale is known, the expected value of future observations are not of values ​​depends observed before. This does not even necessarily the Markov property that the distribution depends only on. For example, the scattering of Martingale also depend on field observations.

The information that is known at the time on the stochastic process can be given generally by a filtration. A filtration is a sequence of σ - algebras, which is sorted in ascending order, that is valid for all. The process is called integrable martingale with respect to the filtration if:

  • For all measurable with respect to is ( one says " the process is adapted to the filtration " ) and
  • For all the Martingalgleichung applies.

The above- considered case of a martingale " par excellence " is included in this definition. One chooses to for the σ - algebra generated by.

Generalization to general index sets

Be a stochastic process on a probability space with an arbitrary ordered index set.

Is called a martingale with respect to a filtration if:

  • For all is measurable with respect to,
  • Applies to each, and
  • For all with valid (P -almost surely ).

The discrete-time case is in this general definition contained, because of all follows inductively for all with. Pay close attention to the case of arbitrary non-negative time points as index set.

Sub - and supermartingale

As a submartingale is called an adapted and integrable stochastic process, which tends to increase in contrast to the martingale:

Accordingly, a supermartingale is an adapted and integrable stochastic process, which tends to be:

Motivating Example

The concept of martingale can be regarded as formalization and generalization of fair gaming. Be to the starting capital of the player. This will be a constant in many cases, but also a random start-up capital is conceivable. The random profit in the first game will be denoted by. It can be positive, zero or negative ( that is, a loss ). The player's capital after the first game and is generally after th game

If the profit referred to in th game. In a fair gamble, the expected value of each prize is equal to zero, that is, it applies to all.

The game play will now be observed up to time including, that is, the capital levels are known. Now, if the profit in the next, ie in the -th, game is independent of the run of play, then the expected total capital is calculated after the next game, taking into account all available information using the calculation rules for conditional expectations to

Thus we have shown that, as a martingale can model the capital of a player who participates in a fair gamble.

In real Gücksspielen, such as in roulette, but it is because of the advantage of the bank expected profit at every game generally negative, ie. Then, then, analogous to the above statement

From the player's point of view, in this case a supermartingale ( mnemonic: " supermartingales are great for the casino ").

Examples of continuous-time martingales

  • A Wiener process is a martingale, as are a Wiener process, the processes and the geometric Brownian motion without drift Martingale.
  • A Poisson process with rate, which is adjusted for its drift, therefore, is a martingale.
  • By the lemma of Itō applies: Each Itō integral ( with limited integrand ) is a martingale. After Itoschen Martingaldarstellungssatz every martingale can be reversed ( even every local martingale ) pose with respect to a filtration generated by a Brownian motion as Ito integral with respect to this same Brownian motion.
  • Each continuous martingale is either of infinite variation or constant.
  • Each stopped martingale is a martingale again.

Quadratic variation and Exponentialmartingal

Is the quadratic variation of a continuous bounded martingale (or a finite exponential moments ) is finite, then the stochastic process

Also a martingale.

Likewise, the so-called Exponentialmartingal of, given by

A martingale.

Origin of the word

The Martingale is a well known since the 18th century strategy in the game of chance in which increases after losing a game of use, is doubled in the simplest case, so that one councils in the hypothetical case of inexhaustible wealth, inexhaustible time, and the non-existence of a maximum use of safe profit.

Since the Martingale was the most famous game system and is, the term was also used as a synonym for " game the system " and so was included in the mathematical literature.

The word " martingale " itself comes from the Provencal and is derived from the French town of Martigues in the Bouches du Rhone on the edge of the Camargue from whose inhabitants were once considered somewhat naive. The Provencal expression jouga a la martegalo means to play as much as very reckless.

The " martingale " said auxiliary reins should also be named after the town of Martigues, this is an optional part of horse equipment that is to prevent the horse from tearing the head up and to rise. That this Reins also called martingale, the pioneers of the martingale was not known - and has nothing to do with the mathematical concept formation.

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