Geodesic
Optimal stopping for absolute maximum of homogeneous diffusion
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2015), pp. 7-13

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The article deals with the optimal stopping problem in case when the reward function depends on the absolute maximum of some homogeneous diffusion. We consider cases of infinite and finite time horizon. In both cases the differential equation for the optimal stopping boundary is obtained. Also, we prove that the maximality principle holds for reward functions which satisfy single-crossing condition. 
@article{VMUMM_2015_5_a1,
     author = {A. A. Kamenov},
     title = {Optimal stopping for absolute maximum of homogeneous diffusion},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {7--13},
     publisher = {mathdoc},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2015_5_a1/}
}
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A. A. Kamenov. Optimal stopping for absolute maximum of homogeneous diffusion. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2015), pp. 7-13. http://geodesic.mathdoc.fr/item/VMUMM_2015_5_a1/