For right continuous martingales and a collection of stopping times, for each $T>0$, $\{X(T\wedge \tau):\tau \in S\}$ is uniformly integrable

Question

Let $X$ be a right continuous $\{\mathscr{F}_t\}$-martingale and let $S$ be the collection of $\{\mathscr{F}_t\}$-stopping times. Show that for each $T>0$, $\{X(T\wedge \tau):\tau \in S\}$ is uniformly integrable.

I thought I should use Optional Sampling Theorem. So first, I have

$$E[X(T\wedge \tau)|\mathscr{F}_\tau]=X(T\wedge \tau).$$

Then using tower property and Jensen's inequality I get

$$E|X(T\wedge \tau)|=E\big|E[X(T\wedge \tau)|\mathscr{F}_\tau]\big|\le E[E[|X(T\wedge \tau)|\big|\mathscr{F}_\tau]]=E|X(T\wedge \tau)|.$$

So this gives me $$E[|X(T\wedge \tau)|\mathscr{F}_\tau]=|X(T\wedge \tau)|.$$

What I need to show is that $$E(|X(T\wedge \tau)|I_{|X(T\wedge \tau)|\ge K})\to 0$$ as $K\to \infty$ for all $\tau \in S$. I'm not sure how I can progress from here. Perhaps I need a different approach. How can I prove this?

Answer 1

Hints:

1. Recall (or prove) the following well-known statement:

   Let $Y \in L^1(\mathbb{P})$ be an integrable random variable on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. Then $$\{\mathbb{E}(Y \mid \mathcal{F}); \mathcal{F} \subseteq \mathcal{A} \, \, \text{$\sigma$-algebra}\}$$ is uniformly integrable.

2. Apply the optional stopping theorem to show that $$\mathbb{E}(X_T \mid \mathcal{F}_{T \wedge \tau}) = X_{T \wedge \tau}$$ for any $\mathcal{F}_t$-stopping time $\tau$.
3. Conclude.