We study the quickest detection problem of a sudden change in the arrival rate
of a Poisson process from a known value to an unknown and unobservable value at an
unknown and unobservable disorder time. Our objective is to design an alarm time
which is adapted to the history of the arrival process and detects the disorder
time as soon as possible.
In previous solvable versions of the Poisson disorder problem, the arrival
rate after the disorder has been assumed a known constant. In reality, however,
we may at most have some prior information on the likely values of the new arrival
rate before the disorder actually happens, and insufficient estimates of the new
rate after the disorder happens. Consequently, we assume in this paper that the
new arrival rate after the disorder is a random variable.
The detection problem is shown to admit a finite-dimensional Markovian
sufficient statistic if the new rate has a discrete distribution with
finitely-many atoms. Furthermore, the detection problem is cast as a
discounted optimal stopping problem with running cost for a finite-dimensional
piecewise-deterministic Markov process.
This optimal stopping problem is studied in detail in the special case
where the new arrival rate has Bernoulli distribution. This is a non-trivial
optimal stopping problem for a two-dimensional piecewise-deterministic Markov
process driven by the same point process. Instead of trying to tackle the free
boundary problem associated with an integro-differential operator, we solve the
problem by constructing an exponentially converging sequence of functions using
the iteration of an appropriately defined single-jump operator.