Geodesic
On the existence of solutions of unbounded optimal stopping problems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 310-319.

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Known conditions of existence of solutions of optimal stopping problems for Markov processes assume that payoff functions are bounded in some sense. In this paper we prove weaker conditions which are applicable to unbounded payoff functions. The results obtained are applied to the optimal stopping problem for a Brownian motion with the payoff function $G(\tau,B_\tau)=|B_\tau|-c/(1-\tau)$. 
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M. V. Zhitlukhin; A. N. Shiryaev. On the existence of solutions of unbounded optimal stopping problems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 310-319. http://geodesic.mathdoc.fr/item/TM_2014_287_a17/

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