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

TL;DR

This paper applies martingale techniques to analyze the gambler's ruin problem for correlated random walks, deriving explicit formulas for ruin probabilities and expected durations, including for delayed and pattern-based scenarios.

Contribution

It introduces a martingale-based method for correlated random walks with delays and pattern-based bets, providing new closed-form solutions for ruin probabilities.

Findings

Derived closed-form ruin probabilities for CRW with and without delays.

Developed a martingale technique applicable to general CRW with delays.

Analyzed ruin probabilities for pattern-based betting scenarios.

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.