On a Variation of Gambler's Ruin Problem

Abstract: Assume that letters (from a finite alphabet) in a text form a Markov chain. We track two distinct words, $U$ and $D$. A gambler gains 1 point for each occurrence of $U$ (including overlapping occurrences) and loses 1 point for each occurrence of $D$ (also including overlapping occurrences). We determine the probability of gaining $A$ points before losing $B$ points, where $A$ and $B$ are integers. Additionally, we find the expected waiting time until one of the two events -- gaining $A$ points or losing $B$ points -- occurs.