By S. Kusuoka, A. Yamazaki

A lot of monetary difficulties can formulated as limited optimizations and equilibration in their solutions.Various mathematical theories were delivering economists with critical machineries for those difficulties coming up in monetary thought. Conversely, mathematicians were motivated through numerous mathematical problems raised via fiscal theories. The sequence is designed to compile these mathematicians who have been heavily drawn to getting new demanding stimuli from fiscal theories with these economists who're looking for powerful mathematical instruments for his or her researchers.

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C) rank Vr{p) = rank Wjr(p, q) if ^Wj^{p, q)\ = Ofor some A G R^+. In the following, we omit the subscript T of the matrices Vjr(p) and WAP). Proof Part (a) is straightforward. We prepare the proofs of Part (b) and (c) by introducing some notations and definitions. We let, for t = 1 , . . , T -f 1, the setJt = {jeJ\^{j)eBt-i}. We give the proof under the additional assumption that Jt 7^ 0 for t G [1, T] and JT-\-I = 0 (and we let the reader adapt this proof to the general case). Then the sets Jt (t e [1,T]) define a partition of the set J and we write every z e R'^ 2is z = (zt) with zt G R*^*.

Proof Part (i). 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.

Let N = NU {oc} and let X and Y be topological spaces. 2). ) is lower semicontinuous onXxY. 3) limM(pk{t,Xk,yk) > ^oo{t,x,y). k Let {uk)k be a sequence of measurable mappings from [0,1] to X which converges pointwisely to a measurable mapping UQO- Let (X^) be a sequence of Young measures in ^([0,1]; Y) stably converging to X^ G ^([0,1]; Y). 4) dt. Proof We define a measurable function ip: [0,1] x N x Y ^ [0,-hoo], by xl^{t, k, y) = (pk{t, Uk{t), y). c. (the idea of using N in this kind of semicontinuity result stems from Balder [3]).