Optimal stopping problems for a Brownian motion with disorder on a segment. (English) Zbl 1293.60048

Zhitlukhin, M. V.; Shiryaev, A. N.

Theory Probab. Appl. 58, No. 1, 164-171 (2014); translation from Teor. Veroyatn. Primen. 58, No. 1, 193-200 (2013).

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

MSC:

60G40	Stopping times; optimal stopping problems; gambling theory
60J65	Brownian motion
91G80	Financial applications of other theories

Keywords:

optimal stopping problems; disorder detection problems; Shiryaev-Roberts statistic