Optimum Shewhart tests for Markovian data

We consider the sequential change detection problem for Markovian processes. Adopting a Lorden-like criterion where expected delays are replaced with detection probabilities we end up with a well defined constrained optimization problem which is possible to solve, exactly, in the Markovian case. The optimum scheme turns out to be a modified version of the Shewhart test (known to be optimum in the i.i.d. case) that involves two unknown functions. These functions can be identified by solving a system of two equation which is the result of applying optimal stopping theory and the need for the optimum scheme to be an equalizer rule.