A question about exponential martingale.

Question

We have a>0 and $X_1,X_2,...$ be iid $N(\mu,1)$ random variables. Also $S_0=a$ and $S_n=a+X_1+...+X_n$. Let R be the event that $S_n\leq 0$ for some n. Show that $P(R)\leq e^{-2\mu a}$.
I created a new martingale $Y_n=e^{-2\mu S_n}$, but I'm confused about what should I do next. What stopping time should I consider to make a useful expression for the even R.

Answer 1

As Did suggested you can apply Doob's inequality. If you want to prove it on your own, define a stopping time $T$ by $$T := \min\{0 \leq k \leq n; Y_k \geq 1\}$$

Note that this implies $$\mathbb{P} \left[ \inf_{0 \leq k \leq n} S_k \leq 0 \right] = \mathbb{P} \left[ \max_{0 \leq k \leq n} Y_k \geq 1 \right] = \mathbb{P}[T<\infty] = \mathbb{P}[T<\infty,Y_{T \wedge n} \geq 1]$$

Now apply Markov's inequality and use the fact that $(Y_n)_n$ is a martingale to conclude that $(Y_{T \wedge n})_n$ is a martingale.