How to capture a stopping time: The independent case

摘要

A stopping time τ is controlled by a process {Xt} and we are interested in the detection of its onset using a sequential scheme. The sequential detector is an alternative stopping time T which is based on observations {ξt} that are obtained sequentially. The occurrence of τ initiates a change in the statistical behavior of the pair processes {(Xt, ξt} for times after τ with τ modeling the change-imposing mechanism. We cast the detection problem as two possible constrained optimal stopping problems. We provide the optimum detector for each case when the pair process {(Xt, ξt)} is i.i.d. and τ corresponds to the first entry time of Xt into some fixed and known set A. The resulting optimum schemes accept an interesting form. In particular one of the two detectors is reduced to the well known Shiryaev test (with τ geometrically distributed) when the two processes {Xt} and {ξt} are independent from each other.