Proving uniform integrability

Question

Let $W_t, t \geq 0$ be Brownian motion.

Now, fix $u \in \mathbb{R}$ and define $X_{(k)}^t := \sum_{n=0}^k \frac{(u \sqrt{t})^n}{n!} H_n \left( \frac{W_t}{\sqrt{t}} \right)$

$\textbf{Claim: $\{ X_{(k)}^t \}_{k } $ is uniformly integrable.}$

I would like to prove uniform integrability by using the following fact:

There exists a constant $K$ such that for all $x \in \mathbb{R}$, $$|H_n(x)| \leq K \sqrt{n!} e^{x^2/4} $$

Many thanks! :)

Answer 1

Hints:

1. Show that $$\mathbb{E}\exp \left( \frac{3}{2} \frac{W_t^2}{4t} \right) \ < \infty$$ for any $t>0$. (Use the fact that $W_t \sim p_t(x) \,dx$ for a density $p_t$ which you can write down explicitly.)
2. Use the given estimate for $H_n$ to show that there exists a constant $C=C(t,u)>0$ such that $$|X_{(k)}^t| \leq C \exp \left( \frac{W_t^2}{4t} \right)$$ for all $k \geq 0$.
3. Combine Step 1 and 2 to prove that $$\sup_{k \in \mathbb{N}_0} \mathbb{E}(|X_{(k)}^t|^{3/2}) < \infty.$$
4. Show (or recall) that for any $p>1$ $$\sup_{k} \mathbb{E}(|Y_k|^p) < \infty \implies (Y_k)_k \, \, \text{is uniformly integrable}$$
5. Conclude.