Martingale Approach to Gambler's Ruin Problem for Correlated Random Walks

The gambler’s ruin problem for correlated random walks (CRW), both with and without delays, is addressed using the Optional Stopping Theorem for martingales. We derive closed-form expressions for the ruin probabilities and the expected game duration for CRW with increments $\{1,-1\}$ and for symmetric CRW with increments $\{1,0,-1\}$ (CRW with delays). Additionally, a martingale technique is developed for general CRW with delays. The gambler’s ruin probability for a game involving bets on two arbitrary patterns is also examined.