By S. Kusuoka, A. Yamazaki

A lot of financial difficulties can formulated as restricted optimizations and equilibration in their solutions.Various mathematical theories were offering economists with critical machineries for those difficulties coming up in monetary thought. Conversely, mathematicians were prompted by means of quite a few mathematical problems raised via fiscal theories. The sequence is designed to collect these mathematicians who have been heavily drawn to getting new not easy stimuli from monetary theories with these economists who're looking for powerful mathematical instruments for his or her researchers.

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17]. Step 2: We now prove Scorza Dragoni's theorem. 1 is well known, see [23, Theorem 1] (the projection property in [23, Theorem 1] is ensured by the assumption that X is Suslin). c, that is, the epigraph epi/i = {(t,x,r) G Cle,n x X^ x R; r > f{t,x)} is closed in Is 2 e, n X X-n fi xXn xM. Let Qe = nn^6,n. c. 2 (Semicontinuity theorem for inductive limits of separable metrizable spaces). Assume that X is a Suslin regular topological space. Assume furthermore that X is the inductive limit of an increasing sequence (X^) c>/metrizable Borel subspaces.

To every asset j E J and every node ^' > ^(j) which is not a maturity node^ of j we define the new asset J = 0 , ^ 0 ' which is issued at ^', and has the same payoffs as asset j at every node which succeeds ^'. 1. The retrading of asset j e J at node ^' G P, denoted J = (j, ^')y is the asset issued at ^', that is, ^{j, ^') = ^', and whose flow of payoffs is given by v{p^ ^, 0', CO) = 0^ otherwise. Given the financial structure J^ = {J, {^{j))j£jy V), we associate a new financial structure T — (J, (C(T))T^ j?

For ^ = ^o, we have^(z)(j,^o) = z{j,£,o) for every j G J; from the definitions of W^(p, q) and Wjr{p^ q), we get: - - E ^ . (o^i(o = [W^(P,9)^](0Part fijj. )• H " It is easy to see that the inverse of ip is the mapping tp: U''^^ —> R-'^" defined by Hz)U,0 = z{j,0 - z{j,ri if e / ^0, and V(z)(j,^o) = z{j,^o), if L. Angeloni, B. 2 Relationship between rank V> and rank W> in a multi-period model The next Proposition shows that several properties of the two-date model also hold in the case of short-lived financial structures.