On the Disorder Problem for a Negative Binomial Process

Abstract

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.

Bayesian Quickest Detection of Credit Card Fraud

“…We also see that it is never optimal to stop before Φ α reaches k i , because, before that moment, the integrand in (3.9) remains negative. Indeed, it is well known that there exists a threshold B i ≥ k i such that the optimal stopping time in (3.9) is given by 1, 2, 3, (3.11) which is the first moment at which Φ α exceeds B i (see Bayraktar et al, 2005;Buonaguidi and Muliere, 2015;Gapeev, 2005;Gapeev and Shiryaev, 2013;Johnson and Peskir, 2017;Peskir and Shiryaev, 2002;Shiryaev, 1978). From (3.11) we observe that the optimal threshold is independent of π, the prior probability that fraud occurs immediately, and this is consistent with the general optimal stopping theory (Peskir and Shiryaev, 2006;Shiryaev, 1978).…”

“…We underline that this is the first time that the optimal stopping theory (see, e.g., Peskir and Shiryaev (2006); Shiryaev (1978)) and the results of the aforementioned articles are applied to credit card fraud detection. Further results on the quickest detection for compound Poisson processes were obtained in Bayraktar and Dayanik (2006); Bayraktar et al (2005); Buonaguidi and Muliere (2015); Gapeev (2005); Herberts and Jensen (2004).…”