Advances in Mathematical Economics by S. Kusuoka, A. Yamazaki PDF

By S. Kusuoka, A. Yamazaki

ISBN-10: 4431308989

ISBN-13: 9784431308980

A lot of monetary difficulties can formulated as restricted optimizations and equilibration in their solutions.Various mathematical theories were providing economists with imperative machineries for those difficulties coming up in monetary idea. Conversely, mathematicians were prompted via numerous mathematical problems raised by way of fiscal theories. The sequence is designed to collect these mathematicians who have been heavily attracted to getting new demanding stimuli from fiscal theories with these economists who're looking for potent mathematical instruments for his or her researchers.

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F). Assume, on the contrary, that, for some i G / the set [P*(x) x Z'^]nBjr{p, q) is nonempty, hence contains a couple (x% z*). Clearly the allocation (x, z) belongs to the set 5(A), hence for every i e Lx'^ e X*(A) C int -BL(0, r) and z' G Z\X) C i n t B j ( 0 , r ) . Thus, for t G ]0,1] sufficiently small, x'{t) := x' + t{x' - x') G int5L(0,r) and z'{t) := z' -f t{z' - z') G i n t P j ( 0 , r ) . Clearly (x*(t),z*(t)) belongs to the budget set Bip{p,q) of agent i (for the economy {S,T)) and since x'{t) G X^ := X' H 5L(0,r), z'{t) G Z* := Z^ n Pj(0, r), the couple (x*(t), 2;*(^)) belongs also to the budget set P^(p, q) of agent i (in the economy {Er^^r))- From the definition of P\ we deduce that x'^(t) G P^(x) (since from above x'^{t) := x^ + t{x^ - x*) and x* G P*(x)), hence x*(t) G P^(x) := P*(x) Pi int P L ( 0 , r).

We say that a sequence {(pk) of lower semicontinuous functions lower epiconverges to a lower semicontinuous function ip if, for every sequence (xk) in M converging to x, we have liminffc (pk{xk) > ^{x). We say that {^k) upper epiconverges to ip if, for every y G M, there exists a sequence {yk)k in H converging to y such that limsup;. (fkiVk) ^ ^(2/)- If (^fc) both lower and upper epiconverges to (p, we say that {(pk) epiconverges to (p. These notions are easily extended to normal integrands. ) for every fixed t G [0,1].

Un{'),Vn{')) is uniformly integrable. Then the following holds lim [h{uj,u''{uj),v''{uj))dP{uj) = j\jh{oo,u^{oo),t)dv^{t) dP{u). Let us mention a useful application of the preceding results. 3. Assume that E is a separable Banach space, S is a convex weak* compact subset of the topological dual E', Y and Z are Polish spaces, f: QxYxZ-^K is a Caratheodory mapping such that u)^ sup \\f{u,y,z)\\ (y,z)eYxZ belongs to L^(^,«S,P). Let {v'^) be a sequence of S-measurable mappings from Q. to S which pointwisely converges to an S-measurable mapping v^, let (u^) be a sequence of S-measurable mappings from QtoY which pointwisely converges to an S-measurable mapping v^, and let (z/^) be a sequence of Young measures in ^(^2,5, P ; Z) which stably converges to v^ G 3^(Q, S, P ; Z).

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Advances in Mathematical Economics by S. Kusuoka, A. Yamazaki


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