It is shown that Page's stopping time is optimal for the detection of changes in distributions, in a well defined sense. This work is a generalization of an existing result where it was shown that Page's stopping time is optimal asymptotically.
"Optimal Stopping Times for Detecting Changes in Distributions." Ann. Statist. 14 (4) 1379 - 1387, December, 1986. https://doi.org/10.1214/aos/1176350164