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On multivalued martingales whose values may be unbounded: Martingale selectors and Mosco convergence. (English) Zbl 0746.60051
Two results on the existence of martingale selections for a multivalued martingale are proved using classical properties of the projective limit of a sequence of subsets. Also, some further properties of the martingale selections are established. Finally some applications are given. The results may have applications in stochastic optimization or control.
Reviewer: A.Gut (Uppsala)

MSC:
60G48 Generalizations of martingales
60D05 Geometric probability and stochastic geometry
60G42 Martingales with discrete parameter
52A22 Random convex sets and integral geometry (aspects of convex geometry)
28B05 Vector-valued set functions, measures and integrals
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
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[1] Assani, I.; Klei, H.A., Parties décomposables compactes de L1(E), C. R. acad. sci. Paris, ser. 1, 294, 533-536, (1982) · Zbl 0489.46028
[2] Attouch, H., (), Applicable Mathematics Series
[3] Bagchi, S., On a.s. convergence of classes of multivalued asymptotic martingales, Ann. inst. H. Poincaré probab. statist., 21, n^{o} 4, 313-321, (1985) · Zbl 0577.60010
[4] Beer, G., Support and distance functionals for convex sets, Numer. funct. anal. optim., 10, Nos. 1, 2, 15-36, (1989) · Zbl 0696.46010
[5] Bismut, J.M., Intégrales convexes et probabilités, J. math. anal. appl., 42, 639-673, (1973) · Zbl 0268.60003
[6] Bourbaki, N., (), Chaps. 1 à 4
[7] Castaing, C., Le théorème de Dunford-Pettis généralisé, C. R. acad. sci. Paris ser. A, 268, 327-329, (1969) · Zbl 0184.40203
[8] Castaing, C., Quelques résultats de convergence des suites adaptées, () · Zbl 0725.60038
[9] Castaing, C.; Touzani, A.; Valadier, M., Théorème de hoffmann-Jørgensen et application aux amarts multivoques, () · Zbl 0628.60059
[10] Castaing, C.; Valadier, M., (), Lecture Notes in Math.
[11] Choukairi-Dini, A., M-convergence et régularité des martingales multivoques, epi-martingales, (), 49-71, 1990 · Zbl 0733.60068
[12] Choukairi-Dini, A., ()
[13] Choukairi-Dini, A., M-convergence des martingales. applications aux amarts uniformes et aux intégrandes, () · Zbl 0733.60064
[14] Choukairi-Dini, A., M-convergence des martingales (asymptotiques) multivoques. epi-martingales, C. R. acad. sci. Paris ser. I, 309, 880-892, (1989) · Zbl 0704.60043
[15] Coste, A., Sur LES multimesures à valeurs fermées bornées d’un espace de Banach, C. R. acad. sci. Paris, 280, 567-570, (1975) · Zbl 0295.46069
[16] Coste, A., Sur LES martingales multivoques, C. R. acad. sci. Paris, 290, 953-956, (1980) · Zbl 0446.60030
[17] Daures, J.P., Convergence presque sure des martingales multivoques à valeurs dans LES convexes compacts d’un espace de Fréchet séparable, C. R. acad. sci. Paris, 274, 1735-1738, (1972) · Zbl 0244.60006
[18] Daures, J.P., Version multivoque du théorème de Doob, C. R. acad. sci. Paris, 275, 527-530, (1972) · Zbl 0244.60007
[19] Daures, J.P., Version multivoque du théorème de Doob, Ann. inst. H. Poincaré, 9, n^{o} 2, 167-176, (1973) · Zbl 0263.60018
[20] Diestel, J.; Uhl, J.J., (), Math. Surveys
[21] Egghe, L., ()
[22] Hess, C., Loi de probabilité et indépendance des ensembles aléatoires à valeurs fermées dans un espace de Banach, () · Zbl 0539.60010
[23] Hess, C., Quelques théorèmes limites pour des ensembles aléatoires bornés ou non, ()
[24] Hess, C., Mesurabilité, convergence et approximation des multifonctions à valeurs dans un e.l.c.s., () · Zbl 0622.28010
[25] Hess, C., Quelques résultats sur la mesurabilité des multifonctions à valeurs dans un espace métrique, () · Zbl 0728.28008
[26] Hess, C., Measurability and integrability of the weak upper limit of a sequence of multifunctions, (), 226-249, No. 1 · Zbl 0748.47046
[27] Hiai, F., Multivalued conditional expectations, multivalued Radon-Nikodym theorems, integral representation of additive operators and multivalued strong law of large numbers, ()
[28] Hiai, F., Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. amer. math. soc., 291, No. 2, (1985) · Zbl 0583.60007
[29] Hiai, F.; Umegaki, H., Integrals, conditional expectations and martingales of multivalued functions, J. multivariate anal., 7, 149-182, (1977) · Zbl 0368.60006
[30] Klei, H.A., Faible compacité des sélections Bochner-intégrables de multi-applications, Thesis, (1985), Paris
[31] Klei, H.A., A compactness criterion in L1(E) and Radon-Nikodym theorems for multimeasures, Bull. sci. math. (2e), 112, 305-324, (1988) · Zbl 0821.46049
[32] Luu, D.Q., Multivalued quasi-martingales and uniform amarts, Acta math. Vietnam, 7, No. 2, 3-25, (1982) · Zbl 0561.60057
[33] Luu, D.Q., On convergence of multivalued asymptotic martingales, ()
[34] Luu, D.Q., Representation of multivalued regular uniform amarts, ()
[35] Luu, D.Q., Applications of set-valued Radon-Nikodym theorems to convergence of multivalued L1-amarts, Math. scand., 54, 101-113, (1984) · Zbl 0562.60057
[36] Mosco, U., Convergence of convex sets and of solutions of variational inequalities, Adv. in math., 3, 510-585, (1969) · Zbl 0192.49101
[37] Neveu, J., Martingales à temps discret, (1972), Masson et Cie Paris
[38] Neveu, J., Convergence presque sûre de martingales multivoques, Ann. inst. H. Poincaré B, 8, n^{o} 4, 1-7, (1972) · Zbl 0235.60010
[39] Neveu, J., ()
[40] Rao, K.M., Quasi-martingales, Math. scand., 24, 79-92, (1969) · Zbl 0193.45502
[41] Rockafellar, R.T., Integral functionals, normal integrands, and measurable selections, (), Lecture Notes in Math. · Zbl 0374.49001
[42] Serpollier, N., Thèse de 3ème cycle, (1979), Université Pierre et Marie Curie Paris
[43] Thiam, D.S., Thèse, (1976), Paris
[44] Van Cutsem, B., Martingales de multiapplications à valeurs convexes compactes, C. R. acad. sci. Paris, 269, 429-432, (1969) · Zbl 0183.45902
[45] Van Cutsem, B., Eléments aléatoires à valeurs convexes compactes, Thesis, (1971), Grenoble
[46] Van Cutsem, B., Martingales de convexes fermés aléatoires en dimension finie, Ann. inst. H. Poincaré B, 8, No. 4, 365-385, (1972) · Zbl 0252.60022
[47] Wets, R., Convergence of convex functions, variational inequalities and convex optimization problems, (), 375-403
[48] Wijsman, R.A., Convergence of sequences of convex sets, cones and functions, Bull. amer. math. soc., 70, 186-188, (1964) · Zbl 0121.39001
[49] Wijsman, R.A., Convergence of sequences of convex sets… II, Trans. amer. math. soc., 123, 32-45, (1966) · Zbl 0146.18204
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