By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to collect these mathematicians who're heavily drawn to getting new hard stimuli from monetary theories with these economists who're looking potent mathematical instruments for his or her study. loads of fiscal difficulties might be formulated as limited optimizations and equilibration in their recommendations. quite a few mathematical theories were delivering economists with quintessential machineries for those difficulties bobbing up in fiscal concept. Conversely, mathematicians were motivated by means of a variety of mathematical problems raised by means of monetary theories

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**Extra info for Advances in mathematical economics. Vol.19**

**Sample text**

RC with the properties Theorem 19. /j˛ ˇj, for all n 2 N, for all ! /, for all n 2 N, for all ! :; ˛// is a martingale. Then there exists an L12C C -bounded RC -valued C -Lipschitz integrand X1 W Œ0; 1 ! s. Proof. s. 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.

Springer, Tokyo, pp 11–44 6. Castaing C, Raynaud de Fitte P (2013) Law of large numbers and ergodic theorem for convex weak star compact valued Gelfand-integrable mappings. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 17. Springer, Tokyo, pp 1–37 7. Castaing C, Raynaud de Fitte P, Valadier M (2004) Young measures on topological spaces. With applications in control theory and probability theory. Mathematics and its applications, vol 571. Kluwer Academic, Dordrecht 8. Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions.

Uk /k converges to u uniformly on any compact set. 15 See Samuelson [14] or Hurwicz and Uzawa [7]. 16 Note that under these conditions, %; %k must satisfy completeness and continuity. 46 Y. Hosoya Proof. x; v/ 2 Limsupk %k . x; v/ and any k0 2 N, there exists k k0 such that %k \U ¤ ;. x; v/k Ä m1 . xm ; vm /j ! 0. xm ; vm / ! xm ; vm / ! x; v/. x; v/ 1, and thus Limsupk %k %. x; v/ 2%. x; v/ 1. 1 /v/ > 1 for any m. x; v/. 1 m1 /v/ 2 U . 1 m1 /v/ ! 1 m1 /v/ 2%k for any k k0 . Hence, we have % Liminfk %k .