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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 $n$.

The example concludes this because the conditional expectation of the discounted time-3 stock price, given information at time 2, equals the discounted time-2 stock price. Specifically, $\mathbb{E}_2\left[\frac{S_3}{(1+r)^3}\right] = 82.64$ and $\frac{S_2}{(1+r)^2} = 82.64$, satisfying the martingale condition $M_n = \mathbb{E}_n[M_{n+1}]$.

Conditions: The expectation is taken under the risk-neutral probability $p = 2/3$.; Discounting uses the interest rate $r = 0.1$.

The example concludes this because the conditional expectation of the discounted time-3 stock price, given information at time 2, equals the discounted time-2 stock price. Specifically, $\mathbb{E}_2\left[\frac{S_3}{(1+r)^3}\right] = 82.64$ and $\frac{S_2}{(1+r)^2} = 82.64$, satisfying the martingale condition $M_n = \mathbb{E}_n[M_{n+1}]$.

Conditions: The expectation is taken under the risk-neutral probability $p = 2/3$.; Discounting uses the interest rate $r = 0.1$.