Examples of markovian moments $\tau$ and $\sigma$ : 1)$EX_\tau = EX_0$ 2) $EX_\tau != EX_0$

Question

Examples of markovian moments $\tau$ and $\sigma$ : 1)$EX_\tau = EX_0$ 2) $EX_\tau \ne EX_0$

Where $X_n$, $n>=0$ martingale

Answer 1

It appears "Markovian moments" is indeed another term for stopping times, so here are some examples with the properties you wanted:

If we let $X$ be a simple random walk, i.e. $X_0=0$ and $X_{n+1} = X_n + Y_n$ where $P(Y_n =1)=P(Y_n = -1) = \frac 12$ is a Bernoulli distributed random variable, then $X$ is a martingale. If we take $\tau := \inf\{n : X_n = 1\}$ then $X_\tau = 1$ so $\mathbb{E}[X_\tau] = 1 \ne 0 = \mathbb{E}[X_0]$.

If we take $\tau := \inf\{n : |X_n| = 1\}$ then $\tau = 1$ so $\mathbb{E}[X_\tau] = \mathbb{E}[X_1] = 0 = \mathbb{E}[X_0].$ Note that any constant time is a stopping time, so this could have been done with any constant $\tau$. I chose this one because in general we have that if $\tau_c := \inf\{n : |X_n| = c\}$ then $\mathbb{E}[X_{\tau_c}] = \mathbb{E}[X_0]$, although it takes slightly more work to show.