Finite Population Fraud Risk in Batch Escrow Processing

I develop an exact finite-population model for fraud exposure in batch escrow processing. Modeling a batch of N transactions with exactly K fraudulent trades processed sequentially in a hypergeometric environment, I derive closed-form expressions for the distribution of the first fraud encounter, including the pmf, survival function, hazard rate, expectation, and variance. These results yield interpretable operational risk metrics such as incident probablity after processing m transactions, expected safe run length, and quantile-based checkpoints. I further formulate and solve an optimal stopping problem that balances throughput against expected loss, showing that the optimal policy is a montone trhreshold characterized by a one-step marginal condition. Monte Carlo simulations validate the theoretical results and compare the derived policies to standard baselines.