Editing Martingale (probability theory) (section)

==Submartingales, supermartingales, and relationship to harmonic functions{{anchor|Submartingales and supermartingales}}==

There are two generalizations of a martingale that also include cases when the current observation ''X<sub>n</sub>'' is not necessarily equal to the future conditional expectation ''E''[''X''<sub>''n''+1</sub>&nbsp;|&nbsp;''X''<sub>1</sub>,...,''X<sub>n</sub>''] but instead an upper or lower bound on the conditional expectation. These generalizations reflect the relationship between martingale theory and [[potential theory]], that is, the study of [[harmonic function]]s. Just as a continuous-time martingale satisfies E[''X''<sub>''t''</sub>&nbsp;|&nbsp;{''X''<sub>''τ''</sub>&nbsp;:&nbsp;''τ''&nbsp;≤&nbsp;''s''}]&nbsp;−&nbsp;''X''<sub>''s''</sub>&nbsp;=&nbsp;0&nbsp;∀''s''&nbsp;≤&nbsp;''t'', a harmonic function ''f'' satisfies the [[partial differential equation]] Δ''f''&nbsp;=&nbsp;0 where Δ is the [[Laplace operator|Laplacian operator]]. Given a [[Brownian motion]] process ''W''<sub>''t''</sub> and a harmonic function ''f'', the resulting process ''f''(''W''<sub>''t''</sub>) is also a martingale.
* A discrete-time '''submartingale''' is a sequence <math>X_1,X_2,X_3,\ldots</math> of [[Integrable function|integrable]] random variables satisfying
::<math>\operatorname E[X_{n+1}\mid X_1,\ldots,X_n] \ge X_n.</math>
: Likewise, a continuous-time submartingale satisfies
::<math>\operatorname E[X_t\mid\{X_\tau : \tau \le s\}] \ge X_s \quad \forall s \le t.</math>
:In potential theory, a [[subharmonic function]] ''f'' satisfies  Δ''f''&nbsp;≥&nbsp;0. Any subharmonic function that is bounded above by a harmonic function for all points on the boundary of a ball is bounded above by the harmonic function for all points inside the ball. Similarly, if a submartingale and a martingale have equivalent expectations for a given time, the history of the submartingale tends to be bounded above by the history of the martingale. Roughly speaking, the [[prefix]] "sub-" is consistent because the current observation ''X<sub>n</sub>'' is ''less than'' (or equal to) the conditional expectation ''E''[''X<sub>n</sub>''<sub>+1</sub>&nbsp;|&nbsp;''X''<sub>1</sub>,...,''X<sub>n</sub>'']. Consequently, the current observation provides support ''from below'' the future conditional expectation, and the process tends to increase in future time.
* Analogously, a discrete-time '''supermartingale''' satisfies
::<math>\operatorname E[X_{n+1}\mid X_1,\ldots,X_n] \le X_n.</math>
: Likewise, a continuous-time supermartingale satisfies
::<math>\operatorname E[X_t\mid\{X_\tau : \tau \le s\}] \le X_s \quad \forall s \le t.</math>
:In potential theory, a [[superharmonic function]] ''f'' satisfies Δ''f''&nbsp;≤&nbsp;0. Any superharmonic function that is bounded below by a harmonic function for all points on the boundary of a ball is bounded below by the harmonic function for all points inside the ball. Similarly, if a supermartingale and a martingale have equivalent expectations for a given time, the history of the supermartingale tends to be bounded below by the history of the martingale. Roughly speaking, the prefix "super-" is consistent because the current observation ''X<sub>n</sub>'' is ''greater than'' (or equal to) the conditional expectation ''E''[''X<sub>n</sub>''<sub>+1</sub>&nbsp;|&nbsp;''X''<sub>1</sub>,...,''X<sub>n</sub>'']. Consequently, the current observation provides support ''from above'' the future conditional expectation, and the process tends to decrease in future time.

===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.