martingale and stopping time

861 Views Asked by At

I currently learn martingale and I am confused on martingale with a stopping time.

Dobb's optional stopping says that if $T$ is bounded, $\{X_n\}$ is a martingale, then $E[X_T] = E[X_0]$.

I have two questions:

  1. Stopping time $T$ is a random variable and $X_n$ is also a random variable. But how to understand $X_T$?

  2. $\{X_n\}$ is a martingale so $\{X_n\}$ have the same expectation already. What's the fancy part of Dobb's optional stopping? I mean why it is important.

Thanks in advance.