Geodesic
Optimal stopping problems for a Brownian motion with disorder on~a~segment
Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 1, pp. 193-200

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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.
Keywords: optimal stopping problems; disorder detection problems; Shiryaev–Roberts statistic. 
@article{TVP_2013_58_1_a10,
     author = {M. V. Zhitlukhin and A. N. Shiryaev},
     title = {Optimal stopping problems for a {Brownian} motion with disorder on~a~segment},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {193--200},
     publisher = {mathdoc},
     volume = {58},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2013_58_1_a10/}
}
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M. V. Zhitlukhin; A. N. Shiryaev. Optimal stopping problems for a Brownian motion with disorder on~a~segment. Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 1, pp. 193-200. http://geodesic.mathdoc.fr/item/TVP_2013_58_1_a10/