# Correlated Betting Streaks and Risk of Ruin: A Martingale Analysis

Cite as: Gorak, R. (2026). Correlated Betting Streaks and Risk of Ruin: A Martingale Analysis. Tradicted. https://www.tradicted.com/research/pozdnyakov-martingale-2025/
Paper: Vladimir Pozdnyakov — *Martingale Approach to Gambler's Ruin Problem for Correlated Random Walks*
Published in: arXiv Working Paper (2025)
Original: https://arxiv.org/abs/2501.10302

Key finding: For a fair-coin game betting on HH against TH, the ruin probability formula α = (B - 1/2)/(A + B) gives α = .25 when A = B = 1.

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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. Doubling after a loss stays solvent only with unbounded capital. A [risk of ruin calculator](/tools/risk-of-ruin-calculator/) puts a number on how quickly a bounded account fails.

## FAQ

### What is the gambler's ruin problem for correlated random walks?

A .25 probability of reaching $1 before losing $1 illustrates the gambler's ruin problem for correlated random walks, per Pozdnyakov (2025). The method applies martingales and the Optional Stopping Theorem to two- and three-state Markov chains representing win/loss sequences, extending the classical i.i.d. gambler's ruin problem.

### How do you calculate risk of ruin when trade outcomes are correlated?

A ruin probability of α=(B-1/2)/(A+B) applies to a fair-coin game betting HH against TH, per Pozdnyakov (2025). The paper builds exponential and quadratic martingales to solve ruin probabilities and expected duration for two- and three-state correlated Markov chains, including versions with delays.

### What formula gives the expected time to ruin for a correlated random walk?

E(τ)=AB(1-p)/p+(A+B)(2p-1)/(2p) is the expected steps to ruin, per Pozdnyakov (2025). The formula applies to a symmetric correlated random walk with increments {1,-1} and reduces to the classical i.i.d. result E(τ) = AB at p = 1/2, matching the independent-bet case.

### Does a winning streak change your odds of eventual ruin?

A symmetric correlated random walk with p=q≠1/2 has ruin probability α=(B-1+1/(2(1-p)))/(A+B-2+1/(1-p)), unlike the i.i.d. formula α=B/(A+B), per Pozdnyakov (2025). Serial correlation in the win/loss sequence shifts the true odds of ruin relative to independent-bet models. The boundaries A and B stay fixed; only the correlation structure changes the answer.

## Source

Pozdnyakov, V. (2025). Martingale Approach to Gambler's Ruin Problem for Correlated Random Walks. arXiv Working Paper.

[Read the full paper →](https://arxiv.org/abs/2501.10302)