# A Stopping Theorem for Right-Continuous Submartingales

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

-
Can you show what you have done so far? – Nate Eldredge Nov 25 '11 at 15:23
Can you do the discrete-time version of this? – GEdgar Nov 25 '11 at 16:02
As $\mathcal F_T\wedge t \subset \mathcal F_t$, then $\mathbb E(X_T \wedge t \mid \mathcal F_T \wedge t)=......$ – Zbigniew Sep 8 '13 at 6:37
@Zbigniew By coincidence, I'm doing this problem right now. I have a little question: to show the process is submartingale, we need to prove that $\mathbb{E}[X_{T\wedge t}] < \infty$ for all $t\geq 0$. How do we prove this? – Integral Oct 23 '15 at 1:41