I'm doing an exercise about martingale and stopping time:
To remove ambiguity, I include how my lecture note define a random walk:
Let $(X_i)_{i=1}^\infty$ be an i.i.d sequence of random variables in which $\mathbb P(X_1 = 1)=p$ and $\mathbb P(X_1 = -1)=1-p$. We define a sequence $(S_n)_{n=0}^\infty$ of random variables by $S_0 = s \in \mathbb R$ and $S_{n} = S_0+ \sum_{i=1}^n X_i$. Then $(S_n)_{n=0}^\infty$ is called a random walk.
In my understanding, for $M_T$ and $S_T$ to be well-defined, $T$ must be finite stopping time. However, it seems to me that there is no guarantee the set $\{n \in \mathbb N \mid S_n \in \{a,b\}\}$ is non-empty.
Could you please elaborate on my confusion? Thank you so much!
Best Answer
It is well known that for $p \neq \frac 1 2 $ the random walk is transient $|S_n| \to \infty$ with probability $1$. Hence if $S_0 \in (a,b)$ it is guaranteed that $S_n\in \{a,b\}$ for some finite $n$ with probability $1$.