A diffusion process and its applications to detecting a change in the drift of brownian motion

The classical cusum procedure of Page (1954) and a competitor suggested independently by Shiryayev (1963) and Roberts (1966) for detecting a change in distribution are systematically compared, when the change point v = 0 and when v is large. The specific model considered is that of detecting a change in the drift of Brownian motion, for which diffusion theory yields certain explicit results that seem impossible to compute in discrete time. The Shiryayev-Roberts process turns out to be a very interesting and in some respects surprising diffusion process. Our conclusion for this simple model is that neither of the two procedures is dramatically better than the other. Examples of more complex problems are given for which the Shiryayev-Roberts procedure seems more easily adapted than the Page procedure.