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On randomized stopping times. (English) Zbl 0732.60058

A characterization of the extremal elements of the set of randomized stopping times by the notion of the optimal projection for processes with indexes in \(R_+\cup \{\infty \}\) is given. Based on this characterization an alternative proof is given that the extremal elements of the set of randomized stopping times are the stopping times [cf. 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) and N. Ghoussoub, Séminaire de probabilités XVI, Univ. Strasbourg 1980/81, Lecture Notes Math. 920, 519-543 (1982; Zbl 0493.60005)].

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G57 Random measures
62L15 Optimal stopping in statistics
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References:

[1] BAXTER, J. R., and CHACON, R. V. (1977): “Compactness of Stopping times{”,Z. Wahrs. verw. Gebiete, 40, 169–181.} · Zbl 0349.60048 · doi:10.1007/BF00736045
[2] DALANG, R. C. (1984): “Sur l’arrêt optimal de processus à temps multidimensionnel continu{”,Séminaire de Probabilitiés XVIII. Lect. Notes in Math., 1059, 379–390.}
[3] DELLACHERIE, C., and MEYER, P. A. (1980): “Probabilités et potentiel{”, chp. V–VIII,Publications de l’institut de mathématique de l’Université de Strasbourg, XVIII, Hermann-Paris.}
[4] EDGAR, G. A.; MILLET, A., and SUCHESTON, L. (1981): “On compactness and optimality of stopping times{”,Lect. Notes in Math., 939, 36–61.} · Zbl 0496.60039 · doi:10.1007/BFb0096258
[5] GHOUSSOUB, N. (1982): “An integral representation of randomized probabilities and its applications{”,Lect. Notes in Math., 920, 519–543.} · Zbl 0493.60005 · doi:10.1007/BFb0092814
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