This paper addresses the risk of fraud in credit card transactions by developing a probabilistic model for the quickest detection of illegitimate purchases. Using optimal stopping theory, the goal is to determine the moment, known as disorder or fraud time, at which the continuously monitored process of a consumer's transactions exhibits a disorder due to fraud, in order to return the best trade-off between two sources of cost: on the one hand, the disorder time should be detected as soon as possible to counteract illegal activities and minimize the loss that banks, merchants and consumers suffer; on the other hand, the frequency of false alarms should be minimized to avoid generating adverse effects for cardholders and to limit the operational and process costs for the card issuers. The proposed approach allows us to score consumers' transactions and to determine, in a rigorous, personalized and optimal manner, the threshold with which scores are compared to establish whether a purchase is fraudulent.
We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.
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