Optional stopping theorem

The Optional Sampling Theorem (also Optional Stopping Theorem ) is a going back to Joseph L. Doob probabilistic statement. A popular version of this theorem states that there is no termination strategy in a fair and repetitive game, with which one can improve his overall profit.

Initial situation

One considers a set of possible time points and a basic set of possible events. At any time a σ algebra exists, which is the level of information at that time. Since the information available over time increases, applies to, i.e. is to filtration. In applications, a probability space is available and it is.

At any given moment there is a - measurable random variable, ie, it is an adapted stochastic process before, can stand at the time, for example, for the payment of a game. It is also assumed that a martingale; the defining condition for presses the fairness of the game from: the forecast on the payment at the time of the at present information is precisely the observation made at. In particular, the expectation value is in at the time with the initial expected value match.

A stopping time is a picture with. The idea is to abort the process at the time, which then leads to the result, which is suitable to define. Whether one breaks up at the time, may only depend on the information to present, which explains Asked on Messbarkeitsbedingung.

It raises the question whether one can obtain a better result than by choosing a suitable stopping time. The optional sampling theorem states that this is not the case under appropriate conditions.

Discrete version

Consider a discrete sequence of moments, one can model this by. The discrete version of the Optional Sampling Theorem states that:

  • Are on a filtration and an adapted martingale and is a stopping time with, and so applies

The questions to, technical requirements are particularly important for the realistic case of limited stop times fulfilled (you can not wait forever ).

The stop strategy to always put on red in roulette, starting each time to double up with a euro to use and the first appearance of red cancel, does not fulfill these technical conditions. However, one has here the unrealistic situation of an unbounded stopping time with exponentially growing operations ( at the "end " to win a total of EUR ).

The following strengthening for bounded stopping times is also referred to as the Optional Sampling Theorem:

  • Are on a filtration and an adapted submartingale and are limited stopping times with so applies

This is the so-called σ - algebra of the past. If, specifically, it is safe and it follows and after application of the expectation. In the case of martingales one can apply this argument on, and we obtain the statement of the former theorem for bounded stopping times.

Continuous version

In the continuous-time case, which is modeled by additional technical conditions have to be made which allow it to return the proof to the discrete case. Similar to the discrete case, the following two sentences, which are also referred to as the Optional Sampling Theorem.

  • Are on a filtration and an adapted martingale with right-sided continuous paths and is a stopping time with, and so applies
  • Are on a filtration and an adapted submartingale with right-sided continuous paths and are limited stopping times with so applies.

Swell

  • Albrecht Irle: Financial Mathematics, Teubner -Verlag (2003), ISBN 3-519-12640-0
  • Set ( mathematics)
  • Stochastics
  • Stochastic process
Filtration (mathematics) Martingale (probability theory)
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