I have some troubles in understanding the Optional Stopping Theorem by Doobs.
I have a bit of confusion about the following (Brzezniak, Zastawniak - Basic Stochastic Processes p. 58-59):
Let $\xi_n$ is a martingale and $\tau$ is a stopping time wrt a filtration $\mathcal{F}_{n}$ such that:
1) $\tau < \infty$ a.s. 2) $\xi_\tau$ is integrable 3) $E(\xi_n1_{\tau>n})$ goes to $0$ as n go to $\infty$.
Then $E(\xi_\tau) = E(\xi_1)$.
The proof starts saying
$\xi_\tau = \xi_{\tau\land n}+(\xi_\tau-\xi_n)1_{\tau>n}$
and then, by applying $E(.)$
$E(\xi_\tau) = E(\xi_{\tau\land n})+E(\xi_\tau1_{\tau>n})+E(\xi_n1_{\tau>n})$.
Now, the crucial point is: I do not understand the difference between $\xi_\tau$ and $\xi_{\tau\land n}$. I mean, the second one is for sure the stopped martingale at the stopping time $\tau$ while the first one is the martingale evaluated at the instant $\tau$.
But if:
$\xi_{n}=\eta_1+2\eta_2+...+2^{n-1}\eta_n$ ("game martingale") and $\tau<n$
then aren't $\xi_\tau$ and $\xi_{\tau\land n}$ the same?
The proof goes on saying $E(\xi_{\tau\land n})=E(\xi_1)$ thanks to definition of martingale (the stopped process is a martingale itself, so this is clear).
Thank you :)
$\tau\wedge n=\min\{\tau,n\}$. Notice that $n$ is fix and $\tau$ is a random variable. So $\tau\wedge n$ is the random variable that is $\tau$ if $\tau\leq n$ and $n$ if $\tau>n$. Here, $$\xi_{\tau\wedge n}=\begin{cases}\xi_\tau&\tau\leq n\\ \xi_n&\tau>n\end{cases}.$$