Aumann type set-valued lebesgue integral and representation theorem

International Journal of Computational Intelligence Systems

Volumn 2, Issue 1, 2009, Pages 83-90

  Li, Jungang ^a   Li, Shoumei ^a  

^a BEIJING UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Aumann type integral; Representation theorem; Set valued Lebesgue integral; Set valued stochastic process

Indexed keywords

RANDOM PROCESSES;

AUMANN TYPE INTEGRAL; BASIC RESULTS; COMPACT SUBSETS; EUCLIDEAN SPACES; LEBESGUE INTEGRAL; REPRESENTATION THEOREM; SET-VALUED STOCHASTIC PROCESS; STOCHASTIC DIFFERENTIAL INCLUSIONS;

STOCHASTIC SYSTEMS;

EID: 77951492731     PISSN: 18756891     EISSN: 18756883     Source Type: Journal    
DOI: 10.1080/18756891.2009.9727642     Document Type: Article

References (29)

1. Kisielewicz, M., 1997. Set-valued stochastic integrals and stochastic inclusions. Stoch. Anal. Appl., 15:783–800.
2. Richter, H., 1963. Verallgemeinerung eines in der Statistik ben ötigten Satzes der Masstherie. Math. Annalen, 150:85–90.
3. Kudo, H., 1954. Dependent experiments and sufficient statistics. Nat. Sci. Rept. Ochanomizu Univ., Tokyo, 4:151–163.
4. Aumann, R., 1965. Integrals of set-valued functions. J. Math. Anal. Appl., 12:1–12.
5. Kisielewicz, M., 1993. Set-valued stochastic integrals and stochastic inclusions. Discuss. Math., 13:119–126.
6. Kisielewicz, M., 1991. Differential Inclusions and Optimal Control, New York:Kluwer Academic Publishers.
7. Kisielewicz, M., 1993. Properties of solution set of stochastic inclusions. J. Appl. Math. Stoch. Anal., 6:217–236.
8. Kisielewicz, M., Michta, M., and Motyl, J., 2003. Set-valued approach to stochastic control part I (existence and regularity properties). Dynamic Syst. and Appl., 12:405–432.
9. Kisielewicz, M., Michta, M., and Motyl, J., 2003. Set-valued approach to stochastic control part II (viability and semimartingale issues). Dynamic Syst. and Appl., 12:433–466.
10. Li, J., and Li, S., 2008. Set-valued stochastic Lebesgue integral and representation theorems. International Journal of Computational Intelligence Systems, 1:177–187.
11. Da Prato, G., and Frankowska, H., 1994. A stochastic Filippov theorem. Stoch. Anal. Appl., 12:409–426.
12. Motyl, J., 1998. Existence of solutions of set-valued Ito equation. Bull. Acad. Pol. Sci., 46:419–430.
13. Ogura, Y., 2007. On stochastic differential equations with set coefficients and the Black-Scholes model. in the Proceeding of the 8th International Conference on Intelligent Technologies. 2007, Sydney. pp. 300–304.
14. Zhang, W., Li, S., Wang, Z., and Gao, Y., 2007. An Introduction of Set-Valued Stochastic Processes, Beijing/New York:Science Press.
15. Hiai, F., and Umegaki, H., 1977. Integrals, conditional expectations and martingales of multivalued functions. Jour. Multivar. Anal., 7:149–182.
16. Li, S., Ogura, Y., and Kreinovich, V., 2002. Limit Theorems and Applications of Set-Valued and Fuzzy Set- Valued Random Variables, Kluwer Academic Publishers.
17. Yan, J. A., 2004. Measure Theorey, Science Press.
18. Molchanov, I., 2005. Theory of Random Sets, Springer.
19. Kim, B. K., and Kim, J. H., 1999. Stochastic integrals of setvalued processes and fuzzy processes. J. Math. Anal. Appl., 236:480–502.
20. Aubin, J. P., and Frankowska, H., 1990. Set-Valued Analysis, Birkhauser.
21. Craven, B. D., 1982. Lebesgue Measure and Integral, Pitman.
22. Castaing, C., and Valadier, M., 1977. Convex Analysis and Measurable Multifunctions, Lect. Notes in Math, Vol. 580, Berlin, New York:Springer–Verlag.
23. Hu, S., and Papageorgiou, N. S., 1997. Handbook of Multivalued Analysis, Kluwer Academic Publishers.
24. Jung, E. J., and Kim, J. H., 2003. On set-valued stochastic integrals. Stoch. Anal. Appl., 21:401–418.
25. Klein, E., and Thompson, A. C., 1984. Theory of Correspondences Including Applications to Mathematical Economics, John Wiley & Sons.
26. Li, S., Ogura, Y., Proske, F. N., and Puri, M. L., 2003. Central limit theorems for generalized set-valued random variables. J. Math. Anal. Appl., 285:250–263.
27. Li, S., and Ren, A., 2007. Representation theorems, setvalued and fuzzy set-valued Ito intergal. Fuzzy Sets and Syst., 158:949–962.
28. Papageorgiou, N. S., 1995. On the conditional expectation and convergence properties of random sets. Trans. Amer. Math. Soc., 347:2495–2515.
29. Zhang, J., Li, S., Mitoma, I., and Okazaki, Y., 2009. On setvalued stochastic integrals in an M-type 2 Banach space. J. Math. Anal. Appl., 350:216–233.