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Optimal multiple stopping with random waiting times. (English) Zbl 1303.60032
This paper extends a theory of optimal multiple stopping. It is the optimization problem: $$\max_{(\tau_1\leq\dots\leq\tau_n)}{\mathbb E}(Y_{\tau_1}+\dots+Y_{\tau_n})$$ (see [G. W. Haggstrom, Ann. Math. Stat. 38, 1618–1626 (1967; Zbl 0189.18301)] for the basic formulation) with multiple exercise opportunities. Without any further restrictions, the optimal strategy is to exercise all rights at that same time, namely, at the time when one would exercise for $$n = 1$$. Typically, there are restrictions imposed on the exercise dates (see, e.g., [R. Carmona and N. Touzi, Math. Finance 18, No. 2, 239–268 (2008; Zbl 1133.91499)], i.e., restrictions that between each two exercise times there must be a given constant time interval $$\delta$$). In the article under review, random waiting times $$\delta_1,\dots,\delta_{n-1}$$ are introduced. For dealing with such an extended problem, the stopping times depend on the waiting times values. The theory for such problems is developed.

MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 62L15 Optimal stopping in statistics 60J60 Diffusion processes 91B70 Stochastic models in economics 91G10 Portfolio theory
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