On representation and regularity of continuous parameter multivalued martingales

Wenlong, Dong; Zhenpeng, Wang

doi:10.1090/S0002-9939-98-04726-1

On representation and regularity of continuous parameter multivalued martingales
HTML articles powered by AMS MathViewer

by Dong Wenlong and Wang Zhenpeng PDF

Proc. Amer. Math. Soc. 126 (1998), 1799-1810 Request permission

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

Sitadri Bagchi, On a.s. convergence of classes of multivalued asymptotic martingales, Ann. Inst. H. Poincaré Probab. Statist. 21 (1985), no. 4, 313–321 (English, with French summary). MR 823079
Sitadri N. Bagchi, Characterisations of classes of multivalued processes using Riesz approximations, J. Multivariate Anal. 42 (1992), no. 1, 23–34. MR 1177515, DOI 10.1016/0047-259X(92)90076-R
Gerald Beer, Convergence of continuous linear functionals and their level sets, Arch. Math. (Basel) 52 (1989), no. 5, 482–491. MR 998621, DOI 10.1007/BF01198356
Gerald Beer, Support and distance functionals for convex sets, Numer. Funct. Anal. Optim. 10 (1989), no. 1-2, 15–36. MR 978800, DOI 10.1080/01630568908816288
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
R. V. Chacon and L. Sucheston, On convergence of vector-valued asymptotic martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 1, 55–59. MR 394859, DOI 10.1007/BF00539861
S. D. Chatterji, Vector-valued martingales and their applications, Probability in Banach spaces (Proc. First Internat. Conf., Oberwolfach, 1975) Lecture Notes in Math., Vol. 526, Springer, Berlin, 1976, pp. 33–51. MR 0517905
Bong Dae Choi and Louis Sucheston, Continuous parameter uniform amarts, Probability in Banach spaces, III (Medford, Mass., 1980) Lecture Notes in Math., vol. 860, Springer, Berlin-New York, 1981, pp. 85–98. MR 647956
B. K. Dam, Almost sure convergence of set-valued martingales and submartingales, Acta Math. Hungar. 60 (1992), no. 3-4, 197–205. MR 1177674, DOI 10.1007/BF00051637
J.-P. Daurès, Version multivoque du théorème de Doob, Ann. Inst. H. Poincaré Sect. B (N.S.) 9 (1973), 167–176 (French, with English summary). MR 0378087
Claude Dellacherie and Paul-André Meyer, Probabilities and potential. B, North-Holland Mathematics Studies, vol. 72, North-Holland Publishing Co., Amsterdam, 1982. Theory of martingales; Translated from the French by J. P. Wilson. MR 745449
Leo Egghe, Strong convergence of pramarts in Banach spaces, Canadian J. Math. 33 (1981), no. 2, 357–361. MR 617625, DOI 10.4153/CJM-1981-029-8
Nikos E. Frangos, On regularity of Banach-valued processes, Ann. Probab. 13 (1985), no. 3, 985–990. MR 799433
Nikos E. Frangos, On convergence of vector valued pramarts and subpramarts, Canad. J. Math. 37 (1985), no. 2, 260–270. MR 780624, DOI 10.4153/CJM-1985-016-1
Evarist Giné, Marjorie G. Hahn, and Joel Zinn, Limit theorems for random sets: an application of probability in Banach space results, Probability in Banach spaces, IV (Oberwolfach, 1982) Lecture Notes in Math., vol. 990, Springer, Berlin, 1983, pp. 112–135. MR 707513, DOI 10.1007/BFb0064267
Allan Gut and Klaus D. Schmidt, Amarts and set function processes, Lecture Notes in Mathematics, vol. 1042, Springer-Verlag, Berlin, 1983. MR 727477, DOI 10.1007/BFb0073700
Christian Hess, On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence, J. Multivariate Anal. 39 (1991), no. 1, 175–201. MR 1128679, DOI 10.1016/0047-259X(91)90012-Q
Christian Hess, Convergence of conditional expectations for unbounded random sets, integrands, and integral functionals, Math. Oper. Res. 16 (1991), no. 3, 627–649. MR 1120474, DOI 10.1287/moor.16.3.627
Christian Hess, Sur l’existence de sélections martingales et la convergence des surmartingales multivoques, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 1, 149–154 (French, with English summary). MR 1086522
Fumio Hiai and Hisaharu Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), no. 1, 149–182. MR 507504, DOI 10.1016/0047-259X(77)90037-9
Fumio Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291 (1985), no. 2, 613–627. MR 800254, DOI 10.1090/S0002-9947-1985-0800254-X
Đinh Quang Lu’u, Quelques résultats de représentation des amarts uniformes multivoques dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 2, 63–65 (French, with English summary). MR 777611
Annie Millet and Louis Sucheston, Characterizations of Vitali conditions with overlap in terms of convergence of classes of amarts, Canadian J. Math. 31 (1979), no. 5, 1033–1046. MR 546956, DOI 10.4153/CJM-1979-095-2
Annie Millet and Louis Sucheston, Convergence of classes of amarts indexed by directed sets, Canadian J. Math. 32 (1980), no. 1, 86–125. MR 559789, DOI 10.4153/CJM-1980-009-1
Nikolaos S. Papageorgiou, On the theory of Banach space valued multifunctions. I. Integration and conditional expectation, J. Multivariate Anal. 17 (1985), no. 2, 185–206. MR 808276, DOI 10.1016/0047-259X(85)90078-8
Nikolaos S. Papageorgiou, Convergence theorems for Banach space valued integrable multifunctions, Internat. J. Math. Math. Sci. 10 (1987), no. 3, 433–442. MR 896595, DOI 10.1155/S0161171287000516
Nikolaos S. Papageorgiou, Convergence and representation theorems for set valued random processes, Stochastic Anal. Appl. 7 (1989), no. 2, 187–210. MR 997278, DOI 10.1080/07362998908809176
Nikolaos S. Papageorgiou, Convergence and representation theorems for set valued random processes, J. Math. Anal. Appl. 150 (1990), no. 1, 129–145. MR 1059575, DOI 10.1016/0022-247X(90)90200-Y
Nikolaos S. Papageorgiou, On the conditional expectation and convergence properties of random sets, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2495–2515. MR 1290728, DOI 10.1090/S0002-9947-1995-1290728-9
Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211–226. MR 479398, DOI 10.1016/0022-247X(77)90060-9
Gabriella Salinetti and Roger J.-B. Wets, On the convergence of closed-valued measurable multifunctions, Trans. Amer. Math. Soc. 266 (1981), no. 1, 275–289. MR 613796, DOI 10.1090/S0002-9947-1981-0613796-3
Marek Slaby, Strong convergence of vector-valued pramarts and subpramarts, Probab. Math. Statist. 5 (1985), no. 2, 187–196. MR 816695
Makoto Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), no. 4, 301–309. MR 740641, DOI 10.1016/0021-9045(84)90003-0
Zhen Peng Wang and Xing Hong Xue, On convergence and closedness of multivalued martingales, Trans. Amer. Math. Soc. 341 (1994), no. 2, 807–827. MR 1154544, DOI 10.1090/S0002-9947-1994-1154544-X