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

2025/01/17 by V. A. Pozdnyakov, Pozdnyakov, Vladimir · 1 citation
Economics, Econometrics and Finance · Mathematics · #60G42 #60J10 #FOS: Mathematics #Modeling, Simulation, and Optimization #Probability (math.PR) #Probability and Statistical Research #Sports Analytics and Performance #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2501.10302

openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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