The ruin problem for a simple random walk defined on a Markov chain

Let random variables $X_0,X_1, X_2,\dots$ be connected in a homogeneous Markov chain and take the values $1$ and $-1$. Let $S_0=0$, $S_n = S_{n-1}+X_n$ for $ n\in\mathbb{N}$ and $\tau_{AB} =\min\{k\in\mathbb{N}\colon S_k = A \text{ or } S_k = -B\}$ for $A,B\in\mathbb{N}$. The formulas for the ruin probability ${\mathbf P}(S_{\tau_{AB}}=A)$ and the generating function ${\mathbf E}( z^{\tau_{AB}}\mid S_{\tau_{AB}}=A)$ were known earlier. The paper presents their new proof, based on martingale theory and Doob's halting theorem.