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

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.


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