Differentiating random upper semicontinuous functions under the integral sign

Test

Volumn 12, Issue 1, 2003, Pages 241-258

  Rodríguez Muñiz, Luis J ^a   López Díaz, Miguel ^a   Gil, M Ángeles ^a  

^a UNIVERSITY OF OVIEDO   (Spain)

Author keywords

De Blasi differential; Random upper semicontinuous function; S differential; differential

Indexed keywords

EID: 0042532015     PISSN: 11330686     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02595821     Document Type: Article

References (30)

1. ARTSTEIN, Z. (1989). Parametrized integration of multifunctions with applications to control and optimization. SIAM Journal of Control and Optimization, 27:1369-1380.
2. AUMANN, R. (1965). Integrals of set-valued functions. Journal of Mathematical Analysis and its Applications, 12:1-12.
3. BANKS, H. and JACOBS, M. (1970). A differential calculus for multifunctions. Journal of Mathematical Analysis and its Applications, 29:246-272.
4. BURRILL, C. (1972). Measure, Integration, and Probability. McGraw-Hill Book Co., New York.
5. e[0,1] representation of random upper semi-continuous functions. Proceedings of the American Mathematical Society. Accepted, in press.
6. DE BLASI, F. (1976). On the differentiability of multifunctions. Pacific Journal of Mathematics, 66:67-81.
7. DEBREU, G. (1967). Integration of correspondences. In Proceedings Fifth Berkeley Symposium on Mathematics Statistics and Probability, pp. 351-372.
8. DIAMOND, P. and KLOEDEN, P. (1994). Metric Spaces of Fuzzy Sets. World Sci., New Jersey.
9. DUBOIS, D. and PRADE, H. (1982). Towards fuzzy differential calculus. Part 3: Differentiation. Fuzzy Sets and Systems. 8:225-233.
10. DUBOIS, D. and PRADE, H. (1987). On several definitions of the differential of a fuzzy mapping. Fuzzy Sets and Systems, 24:117-120.
11. FLANDERS, H. (1974). Differentiation under the integral sign. American Mathematical Monthly, 80:615-627. Correction, ibid. 81 (1974), 145.
12. GIL, M., LÓPEZ-DÍAZ, M., and RODRÍGUEZ-MUÑIZ, L. (1998). An improvement of a comparison of experiments in statistical decision problems with fuzzy utilities. IEEE Transactions on Systems, Man, Cybernetics, 28:856-864.
13. HERMES, H. (1968). Calculus of set valued functions and control. Journal of Mathematical Mechanics, 18:47-60.
14. HIAI, F. and UMEGAKI, H. (1977). Integrals, conditional expectations, and martingales of multivalued functions. Journal of Multivariate Analysis, 7:149-182.
15. KALEVA, O. (1990). The cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems, 35:389-396.
16. KLEMENT, E., PURI, M., and RALESCU, D. (1986). Limit theorems for fuzzy random variables. Proceedings of the Royal Society of London, Series A, 407:171-182.
17. KUDO, H. (1954). Dependent experiments and sufficient statistics. Natural Sciences Reports of the Ochanomizu University. 4:151-163.
18. LOOMIS, L. and STERNBERG, S. (1968). Advanced Calculus. Addison-Wesley Publishing Co., Reading-Massachussets.
19. LÓPEZ-DÍAZ. M. and GIL, M. (1998). Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications. Journal of Statistical Planning and Inference, 74:11-29.
20. PURI, M. and RALESCU, D. (1981). Différentielle d'une fonction floue. Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 293:237-239.
21. PURI, M. and RALESCU, D. (1983). Differentials of fuzzy functions. Journal of Mathematical Analysis and its Applications, 91:552-558.
22. PURI, M. and RALESCU, D. (1985). The concept of normality for fuzzy random variables. Ann. Prob., 13:1373-1379.
23. PURI, M. and RALESCU, D. (1986). Fuzzy random variables. Journal of Mathematical Analysis and its Applications, 114:409-422.
24. RADSTRÖM, H. (1952). An embedding theorem for spaces of convex sets. Proceedings of the American Mathematical Society, 3:165-169.
25. RAMAMOHANA RAO, C. (1957). On differentiation under the integral sign. The Mathematics Student, 25:143-146.
26. RODRÍGUEZ MUÑIZ, L. (2001). Análisis diferencial e integral de variables aleatorias difusas. Aplicaciones a la Decisión estadística. Ph.D. thesis, University of Oviedo.
27. RODRÍGUEZ-MUÑIZ, L. (2002). Statistical modelling, analysis and management of fuzzy data. In M. G. C. Bertoluzza and D. Ralescu, eds., Studies in Fuzzines and Soft Compouting, 87, pp. 104-116. American Statistical Association, Physica-Verlag, Heidelberg.
28. RODRÍGUEZ-MUÑIZ, L., LÓPEZ-DÍAZ, M., GIL, M., and RALESCU, D. (2003). The s-diferentiability of fuzzy mappings. Information Sciences, pp. 143-146. Accepted, in press.
29. RODRÍGUEZ-SALINAS, B. (1967). Sobre el paso al limite bajo el signo integral. Revista de la Academia de Ciencias, Madrid, 61:471-499.
30. ROJAS-MEDAR, M., BASSANEZI, R., and ROMÁN-FLORES, H. (1999). A generalization of the Minkowski embedding theorem and applications. Fuzzy Sets and Systems, 102:263-269.