xbeta
Martingale

Definition

A discrete-time martingale is a discrete stochastic process $X_1,X_2,X_3,\dots$ with the following properties:

• $E[\mid X_t\mid ]<\mathrm{\infty }$,
• $E[X_{t+1}\mid X_1,\dots ,X_t]=X_t.$

The first property states that the random variable $X_t$ is integrable while the second says that the conditional expectation of the next observation, given the complete history of realizations of the process up to time $t$, is simply equal to the previous value.

Submartingales and Supermartingales

A submartingale is a sequence of integrable random variables such that $E[X_{t+1}\mid X_1,\dots ,X_t]\ge X_t$. Similarly, a supermartingale satisfies $E[X_{t+1}\mid X_1,\dots ,X_t]\le X_t$.

A martingale is therefore both a submartingale and a supermartingale.

References

• Chung, Kai Lai (2001): A Course in Probability Theory, Academic Press, London.

External Links

• Wikipedia:Martingale