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By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to compile these mathematicians who're heavily attracted to getting new not easy stimuli from monetary theories with these economists who're looking potent mathematical instruments for his or her examine. loads of fiscal difficulties should be formulated as limited optimizations and equilibration in their ideas. numerous mathematical theories were delivering economists with integral machineries for those difficulties coming up in financial idea. Conversely, mathematicians were influenced by way of quite a few mathematical problems raised through monetary theories

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F / for some f 2 F , then7 Clearly, f % satisfies the homogeneity of degree zero. , for any x 2 and any neighborhood U of x, there exists y 2 U such that y x. x/ x/ for any x 2 . This result can be shown even if g is g not included in the range of P1 . x/ w D 0. f / for some f 2 F , this condition holds automatically. See Theorem 1 and Proposition 1 of Hosoya [6] for more detailed arguments. g 38 Y. Hosoya f % D f; g 2 and, for any % with f % D f , if it satisfies completeness, p-transitivity, and continuity, then %D%g .

1 m1 /v/ ! 1 m1 /v/ 2%k for any k k0 . Hence, we have % Liminfk %k . Therefore, we have Limk %k D%. ugk /k converges to ug uniformly on any compact set. The next lemma was proved in Hosoya [6]. Thus, we omit the proof of this result. Lemma 2. Let g W ! 0/ D x; where x; v 2 . v v/x. x; v/ D 0 if and only if z is proportional to v. Let a; bŒ be the domain of y. I x; v/. x; v/ 0g is well-defined. x; v/ 2 2 . x; v/ D 0. Lemma 3. gk /k converges to g uniformly on any compact set. ugk /k converges to ug uniformly on any compact set.

France, Paris (1972) 3. Castaing C (2011) Some various convergence results for normal integrands. In: Kusuoka S, Maruyama T (eds) Advances in mathematical economics, vol 15. Springer, Tokyo, pp 1–26 4. Castaing C, Ezzaki F, Lavie M, Saadoune M (2011) Weak star convergence of martingales in a dual space. In: Hudzik H, Lewicki G, Musielak J, Nowak M, Skrzypczak L (eds) Function spaces IX. Banach center publications, vol 92. Polish Academy of Sciences, Institute of Mathematics, Warsaw, pp 45–73 5.

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