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Time consistency means that the expected future value of the martingale, conditioned on the information available today, is always equal to today's value. Formally, $M_n = \mathbb{E}_n[M_m]$ for all $n \le m$.

Conditions: Applies for all pairs of times with $n \le m$.; The expectation is conditional on information at time $n$.

Time consistency means that the expected future value of the martingale, conditioned on the information available today, is always equal to today's value. Formally, $M_n = \mathbb{E}_n[M_m]$ for all $n \le m$.

Conditions: Applies for all pairs of times with $n \le m$.; The 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 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 $n$.