The problem of detecting a change in drift of Brownian motion is considered in the Bayesian framework with the time of change having a (prior) exponential distribution. To the well known problem of finding an optimal stopping rule for "declaring a change," we add the option of continuously controlling the sampling rates--resulting in controlling the variance coefficient of the process. The combined problem of finding an optimal rate function (dynamic sampling) together with an optimal stopping rule is solved and explicit expressions for the quantities of interest are derived. The dynamic sampling procedure is shown to be significantly superior to constant rate sampling. The comparison is most favorable when the expected time until change tends to infinity, where the relative efficiency between the two procedures tends to infinity.
"A Dynamic Sampling Approach for Detecting a Change in Distribution." Ann. Statist. 16 (1) 236 - 253, March, 1988. https://doi.org/10.1214/aos/1176350702