Optimal stopping problems for a Brownian motion with a disorder on a finite interval

Abstract

We consider optimal stopping problems for a Brownian motion and a geometric Brownian motion with a "disorder", assuming that the moment of a disorder is uniformly distributed on a finite interval. Optimal stopping rules are found as the first hitting times of some Markov process (the Shiryaev-Roberts statistic) to time-dependent boundaries, which are characterized by certain Volterra integral equations. The problems considered are related to mathematical finance and can be applied in questions of choosing the optimal time to sell an asset with changing trend.