Stochastic Processes And Their Applications To Change Point Detection Problems

This dissertation addresses the change point detection problem when either the post-change distribution has uncertainty or the post-change distribution is time inhomogeneous. In the case of post-change distribution uncertainty, attention is drawn to the construction of a family of composite stopping times. It is shown that the proposed composite stopping time has third order optimality in the detection problem with Wiener observations and also provides information to distinguish the different values of post-change drift. In the case of post-change distribution uncertainty, a computationally efficient decision rule with low-complexity based on Cumulative Sum (CUSUM) algorithm is also introduced. In the time inhomogeneous case where the post-change drift is a function of time, a two-stage stopping time that contains both CUSUM and sequential probability ratio test (SPRT) procedures is considered, whose expected values are given in explicit form. The two-stage stopping time can provide additional information when comparing with CUSUM stopping times. Moreover, the thesis also addresses the joint distribution of maximum drawdown and maximum drawup for any drifted Brownian motion killed at an independent exponentially distributed random time, and as its application, upper and lower bounds of expected time of the minimum between two correlated CUSUM stopping times are provided, where the correlation coefficient is present explicitly