A note on Bayesian detection of change-points with an expected miss criterion

Summary A process X is observed continuously in time; it behaves like Brownian motion with drift, which changes from zero to a known constant ϑ>0 at some time τ that is not directly observable. It is important to detect this change when it happens, and we attempt to do so by selecting a stopping rule T * that minimizes the “expected miss” E | T −τ| over all stopping rules T . Assuming that τ has an exponential distribution with known parameter λ>0 and is independent of the driving Brownian motion, we show that the optimal rule T * is to declare that the change has occurred, at the first time t for which . Here, with Λ=2λ/ϑ 2 , the constant p * is uniquely determined in (½,1) by the equation .