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On the Novikov-Shiryaev optimal stopping problems in continuous time. (English) Zbl 1111.60024
Summary: A. A. Novikov and A. N. Shiryaev [Theor. Probab. Appl. 49, No. 2, 344–354 (2004); translation from Teor. Veroyatn. Primen. 49, No. 2, 373–382 (2004; Zbl 1092.60018)] gave explicit solutions to a class of optimal stopping problems for random walks based on other similar examples given by D. A. Darling, T. Liggett and H. M. Taylor [Ann. Math. Stat. 43, 1363–1368 (1972; Zbl 0244.60037)]. We give the analogue of their results when the random walk is replaced by a Lévy process. Further we show that the solutions show no contradiction with the conjecture given by L. Alili and A. E. Kyprianou [Ann. Appl. Probab. 15, No. 3, 2062–2080 (2005; Zbl 1083.60034)] that there is smooth pasting at the optimal boundary if and only if the boundary of the stopping region is irregular for the interior of the stopping region.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60G51 Processes with independent increments; Lévy processes
91B70 Stochastic models in economics
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