By Charles Castaing

A lot of financial difficulties should be formulated as restricted optimizations and equilibration in their options. a number of mathematical theories were delivering economists with critical machineries for those difficulties coming up in fiscal idea. Conversely, mathematicians were inspired through a variety of mathematical problems raised by means of monetary theories. The sequence is designed to compile these mathematicians who're heavily drawn to getting new demanding stimuli from monetary theories with these economists who search powerful mathematical instruments for his or her researchers. The editorial board of this sequence includes the next in demand economists and mathematicians: **Managing Editors**: S. Kusuoka (Univ. Tokyo), T. Maruyama (Keio Univ.); **Editors**: R. Anderson (U.C. Berkeley), C. Castaing (Univ. Montpellier), F. H. Clarke (Univ. Lyon I), G. Debreu (U.C. Berkeley), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Okayama Univ.), J.-M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Ohio nation Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), okay. Kamiya (Univ. Tokyo), okay. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ. Tokyo), ok. Nishimura (Kyoto Univ.), M. ok. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ).

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**Example text**

Raynaud de Fitte Proof (a) By hypothesis, there is Ii E M~(§) such that maxvEM~(:Z) A(to, xo, Ii, v) < -17 < O. Also, by the equicontinuity hypothesis, there exists ~ > 0 such that maxvEMHZ)A(t, x, Ii, v) < -17/2 for 0 ::; t - to ::; ~ o+ u c(t) dt ::; ~ so that and Ilx - xoll ::; ~. Take a > 0 such that ft: Iluxo,ll,v(t) - UXO,Il,V(tO) II ::; ft:o+u c(t) dt ::; ~ for all t E [to, to + a] and for all v E K (we also denote by Ii the constant Young measure t f-t lit = Ji). Then, by integrating, l to +A(t, UXO,Il,V(t), Ii, Vt) dt ::; lto+u , mar: U to to v EM+ (Z) A(t, UXO,Il,V(t), Ii, v') dt (TT] <-2 for all v E K and the result follows.

Continuity of the payoff function. Economic Theory, 16, 239-244, 2000 [Kingman and Taylor(1966)] Kingman, J. P. : Introduction to measure and probability. : Equilibrium points in N-person games. Proc. Natl. Acad. Sci. : Basic methods of linear functional analysis. London: Hutchinson 1973 [Royden(l988)] Royden. : Real Analysis, 3rd edn. London: The Macmillan Company, Collier-Macmillan 1988 Adv. Math. Econ. 6, 55--68 (2004) Advances in MATHEMATICAL ECONOMICS ©Springer-Verlag 2004 Recursive methods in probability control Seiichi Iwamoto Department of Economic Engineering, Graduate School of Economics, Kyushu University 27, Hakozaki 6-19-1, Higashiku, Fukuoka 812-8581, Japan Received: April 14,2003 Revised: July 22,2003 JEL classification: C61, D81 Mathematics Subject Classification (2000): 90C39, 90C40, 90A43 Abstract.

L 1 E A such that Fiber product of Young measures, application to a control problem 35 where UX,pl " I (/1,1) is the solution trajectory defined on [T, T + a] associated with b 1(p,1), p,1) b 1 E tl, p,1 E A). (p,2)t(z)] dt. JIZ Js T+U Let us define'Y E tl by setting, foraB p, E A: 'Y(p,)t = ,/1 (p,)t fort E [T, a] and 'Y(p,)t = ,/2 (p,)t for t E [T + a, 1]. Coming back to the definition of H J (T, x) HJ(T, x) ::; sup ILEA = 11 [JIZf [Jsf 1 [1[1 T + J(t, UX,P,'Y(p) , s, z) dp,t(s)] d-::y(p,)t(z)] dt T u + J(t, UX,ILI"I(IL I ), s, z) dp,;(s)] d,/l(p,lMz)] dt 11 [f [ f J(t, Vx,p2,,2(1L)' s, z) dP,Z(s)] d,/2(p,2)t(Z)] dt Jz Js T+U ::; KJ(T,X) + 2E.