Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Expressing the following mathematically:

A sequence of random variables $X_0, X_1, \dots$ with finite means such that the conditional expectation of $X_{n+1}$ given $X_0, X_1, X_2, \dots, X_n$ is equal to $X_n$, i.e., $$\mathbb{E}[X_{n+1}\mid X_0, X_1, \dots, X_n] = X_n.$$

A one-dimensional random walk with steps equally likely in either direction $(p=q=\frac12)$ is an example of a martingale.

See these lecture notes for more information.

history | excerpt history