Sequential testing and disorder problems in stochastic diffusion models.
Sequential hypothesis testing and disorder or on-line change-point detection problems form. an important class of optimal stopping problems with partial information. The problem of sequential testing of two simple hypotheses about the dynamics of an observed stochastic process is to determine the time when the observations should be stopped and one of the hypotheses should be accepted. This time is sought to be optimal in the sense of minimal error probabilities and average observation time. The problem of disorder or online change-point detection is to find a time of alarm that should be sounded to indicate a change in probability characteristics of the observed process. This time is sought to be as close as possible to the unknown and unobservable (random) time of change in the sense of minimal probability of a false alarm and average time delay. It is shown that the optimal stopping times in these problems are the first times when the related sufficient statistic processes (weighted likelihood ratios) exit certain continuation regions restricted by the stopping boundaries. The explicit expressions for the corresponding risk functions and the boundaries are derived by means of reducing the initial optimal stopping problems to the associated free-boundary problems and then verifying the candidate solutions using martingale methods. After these problems for observed sequences of independent random variables and Wiener and compound Poisson process with changing probability characteristics were solved, the solution of these problems for observed general diffusion processes and with nonlinear delay penalties became the next challenge.
Literature: 1. Shiryaev A.N.: Optimal Stopping Rules. Springer, 1978 (Chapter IV). 2. Peskir G., Shiryaev A.N.: Optimal Stopping and Free-Boundary Problems. Birkhäuser, 2006 (Chapter VI). 3. Gapeev P. V., Shiryaev A. N. (2011). On the sequential testing problem for some diffusion processes. Stochastics: An International Journal of Probability and Stochastic Processes 83(4–6) (519–535).