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The constant expectation property states that the unconditional expectation of a martingale does not change over time and remains equal to its initial value. Formally, $\mathbb{E}[M_n] = M_0$ for all $n$.

Conditions: Applies for all times $n$ in the discrete-time setting.; This is the unconditional expectation, not the conditional one.

The constant expectation property states that the unconditional expectation of a martingale does not change over time and remains equal to its initial value. Formally, $\mathbb{E}[M_n] = M_0$ for all $n$.

Conditions: Applies for all times $n$ in the discrete-time setting.; This is the unconditional expectation, not the conditional one.

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 $n$.