## On the disorder problem for a negative binomial process.(English)Zbl 1328.60102

Summary: 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 $$p \in (0, 1)$$. 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.

### MSC:

 60G40 Stopping times; optimal stopping problems; gambling theory 62C10 Bayesian problems; characterization of Bayes procedures 62L10 Sequential statistical analysis 35R09 Integro-partial differential equations 35R35 Free boundary problems for PDEs
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