Editing Martingale (probability theory) (section)

===Examples of submartingales and supermartingales===
* Every martingale is also a submartingale and a supermartingale.  Conversely, any stochastic process that is ''both'' a submartingale and a supermartingale is a martingale.
* Consider again the gambler who wins $1 when a coin comes up heads and loses $1 when the coin comes up tails.  Suppose now that the coin may be biased, so that it comes up heads with probability ''p''.
** If ''p'' is equal to 1/2, the gambler on average neither wins nor loses money, and the gambler's fortune over time is a martingale.
** If ''p'' is less than 1/2, the gambler loses money on average, and the gambler's fortune over time is a supermartingale.
** If ''p'' is greater than 1/2, the gambler wins money on average, and the gambler's fortune over time is a submartingale.
* A [[convex function]] of a martingale is a submartingale, by [[Jensen's inequality]].  For example, the square of the gambler's fortune in the fair coin game is a submartingale (which also follows from the fact that ''X<sub>n</sub>''<sup>2</sup>&nbsp;−&nbsp;''n'' is a martingale).  Similarly, a [[concave function]] of a martingale is a supermartingale.