Geodesic
Two-sided disorder problem for a~Brownian motion in a~Bayesian setting
Informatics and Automation, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 211-233

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A two-sided disorder problem for a Brownian motion in a Bayesian setting is considered. It is shown how to reduce this problem to the standard optimal stopping problem for a posterior probability process. Qualitative properties of a solution are analyzed; namely, the concavity, continuity, and the smooth-fit principle for the risk function are proved. Optimal stopping boundaries are characterized as a unique solution to some integral equation. 
@article{TRSPY_2014_287_a11,
     author = {A. A. Muravlev and A. N. Shiryaev},
     title = {Two-sided disorder problem for {a~Brownian} motion in {a~Bayesian} setting},
     journal = {Informatics and Automation},
     pages = {211--233},
     publisher = {mathdoc},
     volume = {287},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2014_287_a11/}
}
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A. A. Muravlev; A. N. Shiryaev. Two-sided disorder problem for a~Brownian motion in a~Bayesian setting. Informatics and Automation, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 211-233. http://geodesic.mathdoc.fr/item/TRSPY_2014_287_a11/