Let $\sigma \leq \tau$ be two random times that are no stopping times. I want to create a simple example that shows that for these random times $\mathbb{E}[M_\tau \mid \mathcal{F}_\sigma] = M_\sigma$ cannot hold.
I am aware of the fact that optional stopping does not work for all stopping times. However, all of this has to do with stopping times. I am a bit confused by the random times. Can anybody provide a clear example?
Take a standars symmetric random walk. $P(X_i=1)=P(X_i=-1)=\frac{1}{2}$. Then $S_n = \sum_{i=1}^n X_k$ is a martingale.
Take $\sigma = \{n\geq 0, S_n = \alpha\}$ which is a stopping time. And take $\tau = \sigma +1$, which is not a stopping time. If \begin{align} \mathbb{E}[S_\tau \mid \mathcal{F}_\sigma] = S_\sigma. \end{align} This would imply that \begin{align} \mathbb{E}[S_\tau] = \mathbb{E}[S_\sigma] = \ldots = \mathbb{E}[S_0] = 0. \end{align} However, \begin{align} \mathbb{E}[S_\tau] = \frac{1}{2}(\alpha -1)+\frac{1}{2}(\alpha +1) = \alpha \neq 0 = \mathbb{E}[S_\sigma]. \end{align}