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On randomized tactics and optimal stopping in the plane. (English) Zbl 0579.60034

This paper considers an optimal stopping problem for two-dimensional stochastic processes \((X_ z)\), indexed by \(N^ 2\) or by \(R^ 2\). The approach generalizes that of J. M. Bismut [Z. Wahrscheinlichkeitstheor. Verw. Geb. 38, 169-198 (1977; Zbl 0336.60070)], N. El Karoui [École d’été de probabilités de Saint-Flour IX-1979, Lect. Notes Math. 876, 74-238 (1981; Zbl 0472.60002)], and G. A. Edgar, A. Millet and L. Sucheston [Martingale theory in harmonic analysis and Banach spaces, Proc. NSF-CBMS Conf., Cleveland/Ohio 1981, Lect. Notes Math. 939, 36-81 (1982; Zbl 0496.60039)] for one-parameter processes, and consists in introducing a set containing the set of stopping points (the set of randomized tactics), showing the existence of our optimal element in the larger set, and then proving that the optimal element can be chosen in the original set of stopping points.
For the two-dimensional problem, the properties of the set of randomized tactics are studied, an integral representation of randomized tactics is established, and a necessary condition for the existence of an optimal stopping point is given.
Reviewer: J.Gianini-Pettitt

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G99 Stochastic processes
60G20 Generalized stochastic processes
60G57 Random measures
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