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ISSN 1088-6826(e) ISSN 0002-9939(p)

On representation and regularity of continuous parameter multivalued martingales

Author(s): Dong Wenlong; Wang Zhenpeng
Journal: Proc. Amer. Math. Soc. 126 (1998), 1799-1810.
MSC (1991): Primary 60D05, 60G25, 60G35, 60G48
MathSciNet review: 1485468
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Abstract: In this paper we study multivalued martingales in continuous time. First we show that every multivalued martingale in continuous time can be represented as the closure of a sequence of martingale selections. Then we prove two results concerning the cadlag modifications of continuous time multivalued martingales, in Kuratowski-Mosco convergence and in convergence in the Hausdorff metric respectively.

References:

1.: S.Bagchi, On a.s. convergence of classes of multivalued asymptotic martingales, Ann. Inst. Henri Poincar $\acute {\mathrm{e}}$ 21 (1985), 313-321. MR 87f:60009
2.: S.Bagchi, Characterisations of classes of multivalued processes using Riesz approximations, J.Multivariate Anal. 42 (1992), 23-34. MR 93h:60072
3.: G.Beer, Convergence of continuous linear functionals and their level sets, Arch. Math. 52, (1989), 482-491. MR 90i:46018
4.: G.Beer, Support and distance functionals for convex sets, Number. Funct. Anal. and Optimiz. 10 (1989), 15-36. MR 89m:46031
5.: C.Castaing,M.Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., vol. 580, Springer-Verlag, Berlin, 1977. MR 57:7169
6.: R.V.Chacon,L.Sucheston, On convergence of vector valued asymptotic martingales, Z. Wahrs. verw. Gebiete 33 (1975), 55-59. MR 52:15658
7.: S.Chatterji, Vector values martingales and their applications, Lecture Notes in Math., vol. 526, Spring- er-Verlag, Berlin and New York, 1976, pp. 33-51. MR 58:24529
8.: B.Choi,L.Sucheston, Continuous parameter uniform amarts, Lecture Notes in Math., vol. 860, Sprin- ger-Verlag, Berlin, 1981, pp. 85-98. MR 83f:60075
9.: B.K.Dam, Almost sure convergence of set-valued martingales and submartingales., Acta Math. Hung. 60 (1992), 197-205. MR 94h:60070
10.: J.P.Daures, Version multivoque du th $\acute {\mathrm{e}}$ or $\acute {\mathrm{e}}$ me de Doob, Ann. Inst. Henri Poincar $\acute {\mathrm{e}}$ 9 (1973), 167-176. MR 51:14256
11.: C.Dellacherie,P.A.Meyer, Probabilities and potentials B,Chap. VI, North-Holland, Netherlands, 1982. MR 85e:60001
12.: L.Egghe, Strong convergence of pramarts in Banach spaces, Cand. J. Math. 33 (1981), 357-361. MR 82m:60052
13.: N.E.Frangos, On regularity of Banach-valued processes, Ann. Prob. 13 (3) (1985), 985-990. MR 86k:60011
14.: N.E.Frangos, On convergence of vector valued pramarts and subpramarts, Canad. J. Math. 37 (1985), 260-270. MR 86h:60097
15.: E.Gin $\acute {\mathrm{e}}$ ,M.G.Hahn and J.Zinn, Limit theorems for random sets:An application of probability in Banach space results, Lecture Notes in Math., vol. 990, Springer-Verlag, Berlin and New York, 1983, pp. 112-135. MR 85d:60019
16.: A.Gut,K.D.Schmidt, Amarts and set function processes, Lecture Notes in Math. vol. 1042 Springer- Verlag, Berlin and New York, 1983. MR 85k:60064
17.: C.Hess, On multivalued martingales whose values may be unbounded:martingale selectors and Mosco convergence, J.Multivariate Anal. 39 (1991), 175-201. MR 93e:60090
18.: C.Hess, Convergence of conditional expectations for unbounded random sets, integrals, and integrals functionals, Math. Oper. Res. 16 (3) (1991(August)), 627-649. MR 92h:49012
19.: C.Hess, Existence of martingale selectors and convergence of multivalued supermartingales, C.R.Acad. Sci. Paris S $\acute {\mathrm{e}}$ r. I Math. 312 (1) (1991), 149-154. MR 91m:60083
20.: F.Hiai,H.Umegaki, Integrals,conditional expectations, and martingales of multivalued functions, J.Mul- tivariate Anal. 7 (1977), 149-182. MR 58:22463
21.: F.Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Ameri. Math. Soc. 291 (2) (1985), 613-627. MR 86k:60048
22.: D.Q.Luu, Representations and regularity of multivalued martingales, C.R.Acad. Sci. Paris, Serie 1 300, (2) (1985), 63-65. MR 86h:60096
23.: A.Millet,L.Sucheston, Characterizations of Vitali Conditions with overlap in terms of convergence of amarts, Canad. J. Math. 31 (1979), 1033-1046. MR 80j:60080
24.: A.Millet,L.Sucheston, Convergence of classes of amarts indexed by directed sets, Canad. J. Math. 32 (1980), 86-125. MR 81g:60051
25.: N.S.Papageorgiou, On the theory of Banach space valued multifunctions,1.Integration and conditional expectation, J.Multivariate Anal. 17 (1985), 185-206. MR 87e:28015
26.: N.S.Papageorgiou, Convergence theorems for Banach space valued integrable multifunctions, Inter. J.Math. & Math. Sci. 10, (1987), 433-442. MR 88i:28019
27.: N.S.Papageorgiou, Convergence and representation theorems for set-valued random processes, Stoch. Anal. Appl. 7 (1989), 187-210. MR 91a:60032
28.: N.S.Papageorgiou, Convergence and representation theorems for set-valued random processes, J.Math. Anal. Appl. 150 (1990), 129-145. MR 92a:60117
29.: N.S.Papageorgiou, On the conditional expectation and convergence properties of random sets, Trans. Amer. Math. Soc. 347 (7) (1995), 2495-2515. MR 95i:60037
30.: G.Salinetti,R.Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), 211-226; errata, 66 (1978), 689. MR 57:18828; MR 80a:90114
31.: G.Salinetti,R.Wets, On the convergence of closed-valued measurable multifunctions, Trans. Amer. Math. Soc. 266 (1981), 275-289. MR 82k:28007
32.: M.Slaby, Strong convergence of vector-valued pramarts and subpramarts, Prob. Math. Statist. 5 (1985), 187-190. MR 87c:60042
33.: M.Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Th. 40 (1984), 301-309. MR 86a:41034
34.: Z.P.Wang,X.H.Xue, On convergence and closedness of multivalued martingales, Trans. Amer. Math. Soc. 341 (2) (1994), 807-827. MR 94d:60074

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