I'm studying for work the book: "Stochastic Calculus and Application" by Choen and Elliot 2 ed. In section $4.2$ (pg. 91) it states the discrete version of the Optional Stopping Theorem for bounded Stopping time:
Let $(\Omega,\mathcal{F},P)$ be a filtered probability space with filtration $\{\mathcal{F}_n\}_{n\in\mathbb{N}}$.
Given that $S$ is a stopping time with respect to the filtation It defines the random Variable
$X_S: \omega\mapsto X_{S(\omega)}(\omega) $
and it defines the the stopped process:
$\{X^S_n\}_{n\in\mathbb{N}} = \{X_{\min\{S,n\}}\}_{n\in\mathbb{N}} $.
Suppose that $\{X_n\}_{n\in\mathbb{N}}$ is an
$\{\mathcal{F}_n\}_{n\in\mathbb{N}}-$supermartingale. If $S$ and $T$ are bounded
$\{\mathcal{F}_n\}_{n\in\mathbb{N}}-$stopping times and $S\leq T$ a.s., then $E[X_T |\mathcal{F}_S] \leq X_S$ a.s.
As corollary of the theorem the book states: that if $S$ is a bounded stopping time then:
1) If $X$ is a super-martingale, so is $X^S$.
2) If $X$ is a sub-martingale, so is $X^S$.
3) If $X$ is a martingale, so is $X^S$.
4) If $X$ is a uniformly integrable, so is $X^S$.
I've understood the theorem and I succeed in proving the points 1,2 and 3 but I can't prove point 4 of the corollary. I need some extra hypothesis to start the proof for example that $X$ is a super-martingale (as in Prove that stopped discrete time nonnegative supermartingales are uniformly integrable). AmI missing something?
Best Answer
Apply the definition of uniform intergablity. Suppose $\tau \leq N$. Then $E|X_{\tau \wedge n}|I_{\{|X_{\tau \wedge n}| >\Delta }\}$ $\leq E|X_n|I_{\{ \tau \geq nX_{\tau \wedge n >\Delta }\}}$ $+E(|X_1|+|X_2|+...+|X_N|) I_{\{{\tau <n,Y>\Delta }\}}$ where $Y=|X_1|+|X_2|+...+|X_N|$ The first term does not exceed $E|X_n|I_{\{|X_n| >\Delta\}}$. In the second term note that $P(Y >\Delta ) \to 0$ as $\Delta \to \infty$.