If $S_{n}$ is a simple random walk i.e $X_{k}= +/- 1$ with prob = 0.5
T = inf {n > = 0 |$S_{n}$ = 1} is a stopping time. T is finite almost surely.
.Explain $E[S_{min(n,T)} ]= E [S_{0}]=0$
I know that $E[S_{T}] = 1 $, but I can't see the above , Please help.
Because of the optional stopping theorem for bounded martingales.
In more details, apply the theorem to the bounded martingale $(S_{\inf(t,n)})_{t\ge 0}$.