Editing Martingale (probability theory) (section)

==Definitions==
A basic definition of a [[Discrete-time stochastic process|discrete-time]] martingale is a discrete-time [[stochastic process]] (i.e., a [[sequence]] of [[random variable]]s) ''X''<sub>1</sub>,&nbsp;''X''<sub>2</sub>,&nbsp;''X''<sub>3</sub>,&nbsp;... that satisfies for any time ''n'',

:<math>\mathbf{E} ( \vert X_n \vert )< \infty </math>

:<math>\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n.</math>

That is, the [[conditional expected value]] of the next observation, given all the past observations, is equal to the most recent observation.

===Martingale sequences with respect to another sequence===

More generally, a sequence ''Y''<sub>1</sub>,&nbsp;''Y''<sub>2</sub>,&nbsp;''Y''<sub>3</sub>&nbsp;... is said to be a '''martingale with respect to''' another sequence ''X''<sub>1</sub>,&nbsp;''X''<sub>2</sub>,&nbsp;''X''<sub>3</sub>&nbsp;... if for all ''n''

:<math>\mathbf{E} ( \vert Y_n \vert )< \infty </math>

:<math>\mathbf{E} (Y_{n+1}\mid X_1,\ldots,X_n)=Y_n.</math>

Similarly, a '''[[continuous time|continuous-time]] martingale with respect to''' the [[stochastic process]] ''X<sub>t</sub>'' is a [[stochastic process]] ''Y<sub>t</sub>'' such that for all ''t''

:<math>\mathbf{E} ( \vert Y_t \vert )<\infty </math>

:<math>\mathbf{E} ( Y_{t} \mid \{ X_{\tau}, \tau \leq s \} ) = Y_s\quad \forall s \le t.</math>

This expresses the property that the conditional expectation of an observation at time ''t'', given all the observations up to time <math> s </math>, is equal to the observation at time ''s'' (of course, provided that ''s''&nbsp;≤&nbsp;''t''). The second property implies that <math>Y_n</math> is measurable with respect to <math>X_1 \dots X_n</math>.

===General definition===

In full generality, a [[stochastic process]] <math>Y:T\times\Omega\to S</math> taking values in a [[Banach space]] <math>S</math> with norm <math>\lVert \cdot \rVert_{S}</math> is a '''martingale with respect to a filtration''' <math>\Sigma_*</math> '''and [[probability measure]] <math>\mathbb P</math>''' if
* Σ<sub>∗</sub> is a [[Filtration (probability theory)|filtration]] of the underlying [[probability space]] (Ω,&nbsp;Σ,&nbsp;<math>\mathbb P</math>);
* ''Y'' is [[adapted process|adapted]] to the filtration Σ<sub>∗</sub>, i.e., for each ''t'' in the [[index set]] ''T'', the random variable ''Y<sub>t</sub>'' is a Σ<sub>''t''</sub>-[[measurable function]];
* for each ''t'', ''Y<sub>t</sub>'' lies in the [[Lp space|''L<sup>p</sup>'' space]] ''L''<sup>1</sup>(Ω,&nbsp;Σ<sub>''t''</sub>,&nbsp;<math>\mathbb P</math>;&nbsp;''S''), i.e.
::<math>\mathbf{E}_{\mathbb{P}} (\lVert Y_{t} \rVert_{S}) < + \infty;</math>
* for all ''s'' and ''t'' with ''s''&nbsp;&lt;&nbsp;''t'' and all ''F''&nbsp;∈&nbsp;Σ<sub>''s''</sub>,
::<math>\mathbf{E}_{\mathbb{P}}  \left([Y_t-Y_s]\chi_F\right) =0,</math>
:where ''χ<sub>F</sub>'' denotes the [[indicator function]] of the event ''F''. In Grimmett and Stirzaker's ''Probability and Random Processes'', this last condition is denoted as
::<math>Y_s = \mathbf{E}_{\mathbb{P}} ( Y_t \mid \Sigma_s ),</math>
:which is a general form of [[conditional expectation]].<ref>{{cite book|first1=G. |last1=Grimmett |first2= D.|last2= Stirzaker|title=Probability and Random Processes|edition= 3rd|publisher= Oxford University Press|year= 2001| isbn =978-0-19-857223-7}}</ref>

It is important to note that the property of being a martingale involves both the filtration ''and'' the probability measure (with respect to which the expectations are taken).  It is possible that ''Y'' could be a martingale with respect to one measure but not another one; the [[Girsanov theorem]] offers a way to find a measure with respect to which an [[Itō process]] is a martingale.

In the Banach space setting the conditional expectation is also denoted in operator notation as <math>\mathbf{E}^{\Sigma_s} Y_t</math>.<ref>{{cite book |last=Bogachev |first=Vladimir |title=Gaussian Measures |publisher=American Mathematical Society |pages=372–373 |year=1998 |isbn=978-1470418694}}</ref>