Change -point detection of two -sided alternatives in the Brownian motion model and its connection to the gambler's ruin problem with relative wealth perception

摘要

This thesis addresses the problem of change-point detection in the Brownian motion model with multiple alternatives. Attention is drawn to the 2-CUSUM stopping time and its properties as a means of detecting a two-sided change. It is shown that the 2-CUSUM stopping rule is second-order asymptotically optimal as the frequency of false alarms tends to infinity. The above problem can be related to the gambler's ruin problem in which gamblers make their decisions to quit the game based on the relative change in their wealth. Probabilities of exiting after a pre-specified upward rally in the gambler's wealth (or a pre-specified downward fall) are determined both in the discrete time framework and in the continuous time framework.