The gambler's ruin problem calculates the probability a series of correlated bets drives an account to zero before reaching a target gain. Pozdnyakov (2025), in "Martingale Approach to Gambler's Ruin Problem for Correlated Random Walks," solves this with martingales and the Optional Stopping Theorem. For a fair-coin game betting HH against TH, the ruin probability of reaching $1 before losing $1 is .25. The expected number of bets until ruin equals 2AB+B-A+1 for this game.
What the Study Found
In the HH vs. TH game, the expected number of occurrences of each pattern in n coin flips is (n - 1)/4. The expected waiting time until the first HH is 6 flips, versus 4 flips for the first TH. The expected waiting time until the first occurrence of either pattern is 3 flips. For a symmetric correlated random walk with p = q = 1/2, the ruin probability reduces to α = B/(A + B). The HH vs. TH ruin probability formula, α = (B - 1/2)/(A + B), yields α = .25 when A equals B equals 1.
Methodology
This is a theoretical paper with no empirical dataset, sample size, or time period. The methodology is mathematical derivation using the Optional Stopping Theorem applied to martingales built for two-state and three-state Markov chains representing correlated random walks. Illustrative examples include a fair-coin game betting on the patterns HH and TH, and a three-state Markov chain with transition probabilities 1/2, 1/3, and 1/8.
Key Statistics
| Metric | Finding | Context |
|---|---|---|
| Ruin probability, HH vs. TH game | α = (B - 1/2)/(A + B) | Fair-coin two-pattern game; A, B = dollar targets |
| Expected duration, HH vs. TH game | E(τ) = 2AB + B - A + 1 | Fair-coin two-pattern game |
| Ruin probability, symmetric CRW | α = (B - 1 + 1/(2(1-p))) / (A + B - 2 + 1/(1-p)) | Two-state CRW with increments {1,-1}, p = q |
| Expected duration, symmetric CRW | E(τ) = AB(1-p)/p + (A+B)(2p-1)/(2p) | Two-state CRW with increments {1,-1}, p = q |
| Probability HH occurs before TH | .25 | Fair coin, unconditional |
| Expected wait until first HH | 6 flips | Fair coin |
| Expected wait until first TH | 4 flips | Fair coin |
Why This Matters
Trading streaks are correlated, not independent, so risk-of-ruin math built on i.i.d. assumptions can misstate the true odds of blowing up an account. The martingale technique lets risk managers compute exact ruin probabilities and expected time to ruin under serial correlation. The delay framework can also model position-sizing rules that sometimes skip a bet instead of always wagering. Open questions remain for random walks with more than three states, limiting use in finer-grained risk models.