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A submartingale satisfies $M_n \leq \mathbb{E}_n[M_{n+1}]$, indicating a process that tends to increase in expectation. A supermartingale satisfies $M_n \geq \mathbb{E}_n[M_{n+1}]$, indicating a process that tends to decrease in expectation.

Conditions: Applies to discrete-time stochastic processes.; Expectation is conditional on information at time $n$.

A submartingale satisfies $M_n \leq \mathbb{E}_n[M_{n+1}]$, indicating a process that tends to increase in expectation. A supermartingale satisfies $M_n \geq \mathbb{E}_n[M_{n+1}]$, indicating a process that tends to decrease in expectation.

Conditions: Applies to discrete-time stochastic processes.; Expectation is conditional on information at time $n$.

A sequence of random variables $M_0, M_1, \ldots, M_N$ is a martingale if it is adapted and satisfies the condition $M_n = \mathbb{E}_n[M_{n+1}]$ for all $n = 0, 1, \ldots, N - 1$. This means the expected value of the next variable, given all prior information, is exactly equal to the current variable's value.

Conditions: The sequence must be adapted, meaning $M_n$ depends only on the first $n$ events.; The equality must hold for all time steps from $0$ to $N-1$.

A sequence of random variables $M_0, M_1, \ldots, M_N$ is a martingale if it is adapted and satisfies the condition $M_n = \mathbb{E}_n[M_{n+1}]$ for all $n = 0, 1, \ldots, N - 1$. This means the expected value of the next variable, given all prior information, is exactly equal to the current variable's value.

Conditions: The sequence must be adapted, meaning $M_n$ depends only on the first $n$ events.; The equality must hold for all time steps from $0$ to $N-1$.