By Shigeo Kusuoka, Toru Maruyama

The sequence is designed to compile these mathematicians who're heavily attracted to getting new demanding stimuli from monetary theories with these economists who're looking potent mathematical instruments for his or her study. loads of monetary difficulties may be formulated as restricted optimizations and equilibration in their recommendations. a number of mathematical theories were offering economists with necessary machineries for those difficulties coming up in monetary idea. Conversely, mathematicians were motivated by means of numerous mathematical problems raised via financial theories.

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Econ. 12, pp. 23–69 (2009) 16. , Vol. 580. Springer-Verlag, Berlin and New York (1977) 17. : A functional version of the Birkhoff ergodic theorem for a normal integrand: a variational approach. The Annals of Probability 31, 63–92 (2003) 18. Dudley, R. : Real Analysis And Probability, Chapman-Hall, Mathematics Series. Wadsworth, Inc. (1989) 19. : Weak-star convergence of convex sets. Journal of Convex Analysis 13(3 + 4), 711–719 (2006) 20. : Integrals, conditional expectations and martingales of multivalued functions.

N thus proving (a). 1. 1. Assume that E is reflexive separable Banach lattice, T is a measurable transformation of preserving P , I is the σ -algebra of invariant sets and (Sn )n∈N is a superadditive process in L1E ( , F , P ) satisfying: n |) (i) supn∈N E(|S < ∞. n (ii) For each f ∗ ∈ E ∗ , ( f ∗ ,Sn n )n∈N is uniformly integrable. Then there exists Z∞ ∈ L1E ( , F , P ) verifying: I f ∗ , Z∞ dP = limn→ A f ∗ , Snn dP = A supn∈N f ∗ , E nSn dP ∗. for all A ∈ F and for all f ∗ ∈ E+ (b) Z∞ (ω) ∈ m≥1 cl co{ Snn(ω) : n ≥ m} almost surely.

Most descriptions of the Frobenius theorem1 are very difficult to comprehend because they require deep knowledge of either differential forms or vector fields. Since many economists do not have this knowledge, 1 See, for example, Matsushima [8], Kosinski [7], Hicks [5], Auslander and MacKenzie [2], and Sternberg [11]. S. Kusuoka and T. 1007/978-4-431-54114-1 2, c Springer Japan 2012 39 40 Y. Hosoya they cannot check the proof of the Frobenius theorem. In contrast, to understand our claim and to check our proof, readers only need to know some elementary facts on general topology, linear algebra, implicit and inverse function theorem, and ordinary differential equations (ODEs).