Optimal sequential change-point detection for Markov models under probability criteria over short time intervals

In this paper, we study sequential change detection problems in the Bayesian framework under the probability delay risk according to which the detection goal is to minimize the probability that the detection delay will exceed some admissible delay level. To do this, we represent the detection problem as an optimal stopping problem for some homogeneous Markov chain and, then, for this problem, through the corresponding Snell envelope method, we construct the optimal solution, which is an optimal detection procedure for the probability risk. Moreover, we apply the developed methods to a number of challenging examples in time series frequently arising in applications, such as autoregression, autoregressive GARCH, and epidemic models. Finally, based on Monte Carlo simulation methods, the obtained theoretical results have been numerically illustrated by comparing the developed sequential procedures with the usual Bayesian detection methods.