Editing Martingale (probability theory) (section)

==Martingales and stopping times==
{{Main|Stopping time}}

A [[stopping time]] with respect to a sequence of random variables ''X''<sub>1</sub>,&nbsp;''X''<sub>2</sub>,&nbsp;''X''<sub>3</sub>,&nbsp;... is a random variable τ with the property that for each ''t'', the occurrence or non-occurrence of the event ''τ'' = ''t'' depends only on the values of ''X''<sub>1</sub>,&nbsp;''X''<sub>2</sub>,&nbsp;''X''<sub>3</sub>,&nbsp;...,&nbsp;''X''<sub>''t''</sub>. The intuition behind the definition is that at any particular time ''t'', you can look at the sequence so far and tell if it is time to stop.  An example in real life might be the time at which a gambler leaves the gambling table, which might be a function of their previous winnings (for example, he might leave only when he goes broke), but he can't choose to go or stay based on the outcome of games that haven't been played yet.

In some contexts the concept of ''stopping time'' is defined by requiring only that the occurrence or non-occurrence of the event ''τ''&nbsp;=&nbsp;''t'' is [[statistical independence|probabilistically independent]] of ''X''<sub>''t''&nbsp;+&nbsp;1</sub>,&nbsp;''X''<sub>''t''&nbsp;+&nbsp;2</sub>,&nbsp;... but not that it is completely determined by the history of the process up to time&nbsp;''t''.  That is a weaker condition than the one appearing in the paragraph above, but is strong enough to serve in some of the proofs in which stopping times are used.

One of the basic properties of martingales is that, if <math>(X_t)_{t>0}</math> is a (sub-/super-) martingale and <math>\tau</math> is a stopping time, then the corresponding stopped process <math>(X_t^\tau)_{t>0}</math> defined by <math>X_t^\tau:=X_{\min\{\tau,t\}}</math> is also a (sub-/super-) martingale.

The concept of a stopped martingale leads to a series of important theorems, including, for example, the [[optional stopping theorem]] which states that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.