Reading through the book "Brownian Motion & Stochastic Processes" by Karatzas and Shreve, I found the following problem (problem 3.24, page 20):
Suppose that $ \{ X_t, \mathcal{F}_t \ | \ 0 \leq t < + \infty \}$ is a right-continuous sub-martingale and $ S \leq T $ are stopping times of $ \{ \mathcal{F}_t \} $. Show that:
(i) $\ \ \ \{ X_{T \ \wedge \ t} , \mathcal{F}_t \ | \ 0 \leq t < + \infty \} \ $ is again a sub-martingale
(ii) $ \ \ E[X_{T \ \wedge \ t} \ | \mathcal{F}_S] \geq X_{S \ \wedge \ t}$, $ \forall t $
Does anybody know how to prove that?
Thanks a lot for all your efforts! Regards, Si