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

Vladimir Pozdnyakov

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arXiv · 2501.10302

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

Vladimir Pozdnyakov

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Abstract

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.

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Vladimir Pozdnyakov. 2025-05-31. Martingale Approach to Gambler's Ruin Problem for Correlated Random Walks. https://arxiv.org/abs/2501.10302

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