Prove that expectation of a stopping time is (in)finite

Question

Let $S_n$ be a simple random walk, with $S_0 = 0$ and $S_n = \sum_{i=1} ^{n} X_i$, where $P(X_i = 1) = P(X_i = -1) = \frac{1}{2}$ and the $X_i$ i.i.d.

Define the stopping time $T := \inf \{n \geq 0 : S_n \in \{a,b\} \}$, with $a <0 < b$.

How can I show whether $E[T] < \infty$ ?

I know that random walk in dimension 1 is recurrent, so the hitting times of $a$ and $b$ are finite, but I don't see how I can use this to show $E[T] < \infty$.

Answer 1

Hints:

1. Show that $S_n^2 -n$ is a martingale.
2. Apply the optional stopping theorem to the bounded stopping time $T \wedge n$ to conclude that $$\mathbb{E}(T \wedge n) = \mathbb{E}(S_{n \wedge T}^2)$$ for all $n \in \mathbb{N}$.
3. Show that $|S_{n \wedge T}| \leq M:=\max\{|a|,|b|\}+1$ to conclude that $$\mathbb{E}(T \wedge n) \leq M^2.$$
4. Apply the monotone convergence theorem.