currently I struggle with the following exercise:
- Let $(\epsilon_i)_{i \in \mathbb{N}}$ be i.i.d. random variables with $P(\epsilon_1 = 1) = p = 1-P(\epsilon_1 = -1)$, where $p \in (\frac{1}{2},1)$,
- Denote by $S_n = \sum^n_{k=1} \epsilon_i$ be the Random walk with ''positive'' drift,
- Take $M_n = S_n - (2p-1) n$ to be a $(\sigma(\epsilon_i : i \leq n))_{n \in \mathbb{N}}$, martingale and lastly
- Let $\tau_b = \inf\{{n \geq 0 : S_n = b\}}$ where $b \in \mathbb{Z}$.
Consider the Case that $b > 0$ then I already proved that $M_n$ is indeed an $\mathcal{F}_n$ martingale, moreover I also proved that $E[\tau_b \wedge m] \leq \frac{b}{2p-1}$ and it is left to prove that $E[\tau_b] = \frac{b}{2p-1}$ then I am finished, where the hint says that the optional sampling theorem should be applied.
What I have tried:
- Getting started, $\tau_b$ is an almost sure finite stopping time, since by monotone convergence theorem we can conclude that $\lim_{m \to \infty} E[\tau_b \wedge m] = E[\lim_{m \to \infty} \tau_b \wedge m] \leq \frac{b}{2p-1}$, hence $\tau_b$ is almost surely finite.
- Secondly $E[|M_{\tau_b}|] = E[|b - (2p-1)\tau_b|] \leq C < \infty$, since by the first step $P(\tau_b < \infty) = 1$.
- Now I struggle to prove that $E[|M_n| \mathbf{1}_{\tau > n}] \to 0$ as $n \to \infty$, since $b>0$ one should be able to somehow bound the latter by using that the Random Walk with drift is going to hit $b$ in a finite time but I don't know how to continue.
I am thankful for any hints.