# 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

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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