| Title:
|
Convergence theorems for set-valued conditional expectations (English) |
| Author:
|
Papageorgiou, Nikolaos S. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1993 |
| Pages:
|
97-104 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we prove two convergence theorems for set-valued conditional expectations. The first is a set-valued generalization of Levy's martingale convergence theorem, while the second involves a nonmonotone sequence of sub $\sigma $-fields. (English) |
| Keyword:
|
measurable multifunction |
| Keyword:
|
set-valued conditional expectation |
| Keyword:
|
Levy's theorem |
| Keyword:
|
support function |
| Keyword:
|
Kuratowski-Mosco convergence of sets |
| MSC:
|
28B20 |
| MSC:
|
60D05 |
| MSC:
|
60F99 |
| MSC:
|
60G48 |
| MSC:
|
60G99 |
| idZBL:
|
Zbl 0788.60021 |
| idMR:
|
MR1240208 |
| . |
| Date available:
|
2009-01-08T18:01:32Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118560 |
| . |
| Reference:
|
[1] Alo R., deKorvin A., Roberts R.: The optional sampling theorem for convex set valued martingales.J. Reine Angew. Math. 310 (1979), 1-6. MR 0546661 |
| Reference:
|
[2] Artstein Z., Hart S.: Law of large numbers for random sets and allocation processes.Math. Oper. Res. 6 (1981), 485-492. Zbl 0524.28015, MR 0703091 |
| Reference:
|
[3] Attouch H.: Famille d'opérateurs maximaux monotones et mesurabilité.Ann. Mat. Pura ed Appl. 120 (1979), 35-111. MR 0551062 |
| Reference:
|
[4] Diestel J. Uhl J.: Vector Measures.Math. Surveys, vol. 15, AMS, Providence, RI, 1977. MR 0453964 |
| Reference:
|
[5] Dynkin E., Evstigneev I.: Regular conditional expectations of correspondences.Theory of Prob. and Appl. 21 (1976), 325-338. Zbl 0367.60002, MR 0430204 |
| Reference:
|
[6] Fetter H.: On the continuity of conditional expectations.J. Math. Anal. Appl. 61 (1977), 227-231. Zbl 0415.60003, MR 0455110 |
| Reference:
|
[7] Hanen A., Neveu J.: Atomes conditionels d'un espace de probabilité.Acta Math. Hungarica 17 (1966), 443-449. MR 0205285 |
| Reference:
|
[8] Hess C.: Measurability and integrability of the weak upper limit of a sequence of multifunctions.J. Math. Anal. Appl. 153 (1990), 206-249. Zbl 0748.47046, MR 1080128 |
| Reference:
|
[9] Hiai F.: Radon-Nikodym theorems for set-valued measures.J. Multiv. Anal. 8 (1978), 96-118. Zbl 0384.28006, MR 0583862 |
| Reference:
|
[10] Hiai F., Umegaki H.: Integrals, conditional expectations and martingales of multivalued functions.J. Multiv. Anal. 7 (1977), 149-182. Zbl 0368.60006, MR 0507504 |
| Reference:
|
[11] deKorvin A., Kleyle R.: A convergence theorem for convex set-valued supermartingales.Stoch. Anal. Appl. 3 (1985), 433-445. MR 0808943 |
| Reference:
|
[12] Luu D.Q.: Quelques resultats de representation des amarts uniforms multivoques.C.R. Acad. Su. Paris 300 (1985), 63-63. |
| Reference:
|
[13] Metivier M.: Semimartingales.DeGruyter, Berlin 1982. Zbl 0595.60008, MR 0688144 |
| Reference:
|
[14] Mosco U.: Convergence of convex sets and solutions of variational inequalities.Advances in Math. 3 (1969), 510-585. MR 0298508 |
| Reference:
|
[15] Papageorgiou N.S.: On the efficiency and optimality of allocations II.SIAM J. Control Optim. 24 (1986), 452-479. Zbl 0589.90015, MR 0838050 |
| Reference:
|
[16] Papageorgiou N.S.: Convergence theorem for Banach space valued integrable multifunctions.Intern. J. Math. and Math. Sci. 10 (1987), 433-442. MR 0896595 |
| Reference:
|
[17] Papageorgiou N.S.: On the theory of Banach space valued multifunctions. Part 1: Integration and conditional expectation.J. Multiv. Anal. 17 (1985), 185-206. MR 0808276 |
| Reference:
|
[18] Papageorgiou N.S.: On the theory of Banach space valued multifunctions. Part 2: Set valued martingales and set valued measures.J. Multiv. Anal. 17 (1985), 207-227. MR 0808277 |
| Reference:
|
[19] Papageorgiou N.S.: A convergence theorem for set-valued supermartingales in a separable Banach space.Stoch. Anal. Appl. 5 (1988), 405-422. MR 0912867 |
| Reference:
|
[20] Papageorgiou N.S., Kandilakis D.: Convergence in approximation and nonsmooth analysis.J. Approx. Theory 49 (1987), 41-54. Zbl 0619.41033, MR 0870548 |
| Reference:
|
[21] Salinetti G. Wets R.: On the convergence of sequences of convex sets in finite dimensions.SIAM Review 21 (1979), 18-33. MR 0516381 |
| Reference:
|
[22] Thibault L.: Esperances conditionelles d'integrandes semicontinus.Ann. Inst. H. Poincaré Ser. B 17 (1981), 337-350. MR 0644351 |
| Reference:
|
[23] Wagner D.: Survey of measurable selection theorems.SIAM J. Control Optim. 15 (1977), 859-903. Zbl 0407.28006, MR 0486391 |
| . |
