Minimax optimality in Robust Detection of disorder times in Doubly Stochastic Poisson Process

We consider the minimax quickest detection problem of an unobservable time of proportional
change in the intensity of a doubly-stochastic Poisson process. We seek
a stopping rule that minimizes the robust Lorden criterion, formulated in terms of
the number of events until detection, both for the worst-case delay and the false
alarm constraint. This problem, introduced by Page (1954), has received more attention
in the continuous path framework (for Wiener processes) than for point
processes, where optimality results only concern the Bayesian framework (Peskir-
Shyriaev (2002)). Here, we adopt the robust Lorden criterion in place of the classical
Bayesian point of view, but formulated in terms of the number of events until detection,
both for the worst-case delay and the false alarm constraint. Our setting
concerns doubly stochastic (deaths) point process, and the still so-called cumulative
sums (cusum) strategy defined from the supremum in time of the log-likelihood process.
Both criteria are invariant by time rescaling, which minimizes the impact of
stochastic intensity.
We derive the exact optimality of the cusum stopping rule, conjectured but not
solved for the Poisson case, by using finite variation calculus and elementary martingale
properties to characterize the performance functions of the cusum stopping
rule as solutions of some delayed differential equations that we solve elementary.
The case of detecting a decline in the intensity is easy to study because the performance
functions are continuous (Moustakides (2008)). In case of detecting a a rise,
where the performance functions are not continuous, differential calculus requires
using a discontinuous local time at the discontinuity level, difficult to estimate. The
conjecture was considered proven by the community, but the proof was still lacking
for this reason. Some numerical considerations are provided at the end of the article.

invited by Sylvie Roelly