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.


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