A Banach lattice approach to convergent integrably bounded set-valued martingales and their positive parts

Journal of Mathematical Analysis and Applications

Volumn 342, Issue 2, 2008, Pages 780-797

  Labuschagne, Coenraad C A ^a  

^a UNIVERSITY OF THE WITWATERSRAND   (South Africa)

Author keywords

Banach lattice; Banach space; Hiai Umegaki space; Lebesgue Bochner space; Martingale; Measure space; Measure free martingale; Metric space; Vector lattice

Indexed keywords

EID: 40649088281     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2007.12.013     Document Type: Article

References (37)

1. Barcenas D., and Leiva H. Characterization of extremal controls for infinite dimensional time-varying systems. SIAM J. Control Optim. 40 (2001) 333-347
2. Bochner S. Partial ordering in the theory of martingales. Ann. of Math. 62 (1955) 162-169
3. Bouchitté G., and Fragalà I. Optimality conditions for mass design problems and applications to thin plates. Arch. Ration. Mech. Anal. 184 (2007) 257-284
4. Cascales B., and Rodríguez J. Birkhoff integral for multi-valued functions. special issue dedicated to John Horváth. J. Math. Anal. Appl. 297 (2004) 540-560
5. Castaing C. Sur les multi-application meaurables. Rev. Française Informat. Recherche Opérationnell 1 (1967) 91-126
6. p-spaces. J. Math. Anal. Appl. 323 (2006) 119-130
7. Daurès J.P. Version multivoque du théorèm de Doob. Ann. Inst. H. Poincaré Sect. B (N.S.) 9 (1973) 167-176
8. DeMarr R. A martingale convergence theorem in vector lattices. Canad. J. Math. 18 (1966) 424-432
9. Diestel J., and Uhl J.J. Vector Measures. Math. Surveys vol. 15 (1977), Amer. Math. Soc., Providence, RI
10. Hiai F., and Umegaki H. Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7 (1977) 149-182
11. Krickeberg K. Convergence of martingales with a directed index set. Trans. Amer. Math. Soc. 83 (1956) 313-337
12. Krupa G. Convergence of multivalued mils and pramarts in spaces without the RNP. Studia Sci. Math. Hungar. 40 (2003) 13-31
13. Kuo W.-C., Labuschagne C.C.A., and Watson B.A. Discrete time stochastic processes on Riesz spaces. Indag. Math. (N.S.) 15 (2004) 435-451
14. Kuo W.-C., Labuschagne C.C.A., and Watson B.A. Conditional expectation operators on Riesz spaces. J. Math. Anal. Appl. 303 (2005) 509-521
15. Labuschagne C.C.A., Pinchuck A.L., and van Alten C.J. A vector lattice version of Rådström's embedding theorem. Quaest. Math. 30 (2007) 285-308
16. Li S., and Ogura Y. Convergence of set valued sub- and supermartingales in the Kuratowski-Mosco sense. Ann. Probab. 26 (1998) 1384-1402
17. Li S., Ogura Y., and Kreinovich V. Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables (2002), Kluwer Academic Press, Dordrecht
18. Li S., Ogura Y., Proske F., and Puri M.L. Central limit theorems for generalized set-valued random variables. J. Math. Anal. Appl. 285 (2003) 250-263
19. Luxemburg W.A.J., and Zaanen A.C. Riesz Spaces (1971), North-Holland, Amsterdam
20. Markov S. The mystery of intervals. Reliab. Comput. 7 (2001) 63-65
21. Markov S. On quasivector spaces of convex bodies and zonotopes. Numer. Algorithms 37 (2004) 263-274
22. Martellotti A., and Sambucini A.R. The finitely additive integral of multifunctions with closed and convex values. Z. Anal. Anwendungen 21 (2002) 851-864
23. Michta M., and Motyl J. Set valued Stratonovich integral and Stratonovich type stochastic inclusion. Dynam. Systems Appl. 16 (2007) 141-154
24. Neveu J. Discrete-Parameter Martingales (1975), North-Holland Publishing Co.
25. Ni Z., and Zhang W. Doob decomposition of sub-(super)martingales. Acta Math. Sinica (Chin. Ser.) 35 (1992) 53-62
26. Puri M.L., and Ralescu D.A. Differentials of fuzzy functions. J. Math. Anal. Appl. 91 (1983) 551-558
27. Rådström H. An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3 (1952) 165-169
28. Schaefer H.H. Banach Lattices and Positive Operators (1974), Springer-Verlag, Berlin
29. Stoica G. Martingales in vector lattices. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 34 82 (1990) 357-362
30. Tourky R., and Yannelis N.C. Markets with many more agents than commodities: Aumann's "hidden" assumption. J. Econom. Theory 101 (2001) 189-221
31. Troitsky V. Martingales in Banach lattices. Positivity 9 (2005) 437-456
32. Uhl J.J. Orlicz spaces of finitely additive set functions. Studia Math. 29 (1967) 19-58
33. Uhl J.J. Applications of Radon-Nikodým theorems to martingales of vector valued functions. Bull. Amer. Math. Soc. 75 (1969) 840-842
34. Uhl J.J. Martingales of vector valued set functions. Pacific J. Math. 30 (1969) 533-548
35. Xue X., and Yu J. Periodic solutions for semi-linear evolution inclusions. J. Math. Anal. Appl. 331 (2007) 1246-1262
36. Zaanen A.C. Introduction to Operator Theory in Riesz Space (1997), Springer-Verlag, Berlin
37. Zohry M. Characterization of set-valued conditional expectation. Math. Nachr. 279 (2006) 1376-1382