Mathematical Disproof of the Martingale Strategy on Finite Intervals

Applying geometric progressions to compensate for prior negative outcomes, commonly known as the Martingale method, is one of the most persistent misconceptions in applied statistical analysis. The theoretical convergence of this model to a guaranteed positive outcome relies on two assumptions: infinite capital and the absence of a maximum cap on transaction size. In real-world environments, both assumptions are false, transforming this scheme into a generator of catastrophic ruin risks.

Mathematical modeling shows that with a fixed outcome probability p < 0.5 and a step limit N, the probability of total bankroll exhaustion B increases exponentially. Each subsequent step doubles the exposure of the balance, while the expected net profit of the entire series remains constant and equal to the initial trade minus transaction fees. Thus, the ratio of potential loss to potential reward deteriorates with each step, leading to inevitable marginal collapse over a long sequence.

To protect capital from such systemic failures, data engineers employ dynamic risk-management methods based on current sequence volatility. Instead of exponentially scaling the participation coefficient, a fixed-fraction strategy or an adaptive fractional Kelly criterion is implemented. This keeps portfolio variance within acceptable quadratic deviations and guarantees system stability over a sliding window.