By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to collect these mathematicians who're heavily drawn to getting new demanding stimuli from financial theories with these economists who're looking potent mathematical instruments for his or her examine. loads of monetary difficulties could be formulated as limited optimizations and equilibration in their ideas. quite a few mathematical theories were delivering economists with fundamental machineries for those difficulties bobbing up in financial concept. Conversely, mathematicians were encouraged by means of quite a few mathematical problems raised by way of monetary theories.

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De Blasi FS, Tomassini L (2011) On the strong law of large numbers in spaces of compact sets. J Convex Anal 18(1):285–300 10. Feron R (1976) Ensembles aléatoires flous. C R Acad Sci Paris Sér A-B 282(16):Aiii, A903–A906 11. Fitzpatrick S, Lewis AS (2006) Weak-star convergence of convex sets. J Convex Anal 13(3–4):711–719 12. Hiai F (1984) Strong laws of large numbers for multivalued random variables. In: Multifunctions and integrands (Catania, 1983). Lecture notes in mathematics, vol 1091. Springer, Berlin, pp 160–172 13.

S. /j˛ ˇjj 8! 2 nN 30 C. Castaing et al. ˛ for all ˛; ˇ 2 Q. ; ˛/ for ! ; ˛/ D 0 for ! 2 N . ; ˛/ is a C -Lipschitzean integrand on Q. ; ˛/ Œ0; 1. :; r/ is measurable for any fixed r 2 Œ0; 1 thanks to the measurable projection theorem [8, Theorem III. 23]. ; r 2 Œ0; 1. Now we prove that X1 satisfies the required convergence. jXn˛m ˛m E Fn X1 j; Fn /n2N /m2N is an L1 -bounded submartingale. s. /j˛ ˇj and ˇj for all ˛; ˇ 2 Œ0; 1. By the Lipschitz property of X1 and Jensen’s inequality, we have the estimation ˛ jE Fn X1 ˇ ˛ E Fn X1 j Ä E Fn jX1 ˇ X1 j Ä E Fn C j˛ for all ˛; ˇ 2 Œ0; 1.

Inoue H (1991) A strong law of large numbers for fuzzy random sets. Fuzzy Sets Syst 41(3):285–291 16. Jalby V (1992) Semi-continuité, convergence et approximation des applications vectorielles. loi des grands nombres. Technical report, Université Montpellier II, Laboratoire Analyse Convexe, 34095 Montpellier Cedex 05, France 17. Joo SY, Kim YK, Kwon JS (2006) Strong convergence for weighted sums of fuzzy random sets. Inf Sci 176(8):1086–1099 18. Kruse R (1982) The strong law of large numbers for fuzzy random variables.