By Fettah Akhiat, Charles Castaing, Fatima Ezzaki (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)
Advances in Mathematical Economics is a ebook of the examine heart for Mathematical Economics, which was once based in 1997 as a world clinical organization that goals to advertise learn actions in mathematical economics.
Our e-book used to be introduced to achieve our long term objective of bringing jointly these mathematicians who're heavily drawn to acquiring new tough stimuli from financial theories and people economists who're looking powerful mathematical instruments for his or her research.
The scope of Advances in Mathematical Economics contains, yet isn't constrained to, the subsequent fields: - monetary theories in quite a few fields in response to rigorous mathematical reasoning; - mathematical equipment (e.g., research, algebra, geometry, likelihood) prompted by means of financial theories; - mathematical result of capability relevance to financial concept; - historic examine of mathematical economics.
Authors are requested to boost their unique effects as totally as attainable and likewise to offer a straight forward expository evaluate of the matter less than dialogue. accordingly, we are going to additionally invite articles that can be thought of too lengthy for e-book in journals.
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E/. E/-valued Pettis-integrable multifunctions satisfying: (i) E Fq jXn j < 1 for each n 2 N and each 1 Ä q < n. E/-valued Pettis-integrable martingale, that is, P e-E Fn XnC1 D XRn for all n 2 N. x ; Xn /jdP < 1. Am /. E/-tight for each m 2 N. s. to X1 . Proof. F /. i i / exists. m/ . Am /. m/ 30 F. Akhiat et al. s. Am /. x ; X1 / D hm x a:s: ! x ; X1 / a:s: ! x; X1 / a:s: ! m/ P1 m Put X1 D mD1 X1 1Am . s. to X1 . It is easy to check that X1 is F -measurable. s. to a function in L1 . c0 6,! E/ any F -measurable and scalarly integrable selection of X1 is Pettis-integrable.
Am /. x ; X1 / D hm x a:s: ! x ; X1 / a:s: ! x; X1 / a:s: ! m/ P1 m Put X1 D mD1 X1 1Am . s. to X1 . It is easy to check that X1 is F -measurable. s. to a function in L1 . c0 6,! E/ any F -measurable and scalarly integrable selection of X1 is Pettis-integrable. Accordingly X1 is Pettis-integrable. 2. c0 6,! E/. Xn ; Fn /n2N be an adapted sequence of E-valued Pettis-integrable functions satisfying: (i) E Fq jXn j < 1 for each n 2 N and each 1 Ä q < n. Xn ; Fn /n2N is an E-valued Pettis-integrable martingale, that is, Xn D P e-E Fn XnC1 for all R n 2 N.
His core equivalence theorem is very general in that he assumed a set of very weak conditions on traders’ preferences. In fact, in his We are grateful to Prof. Ezra Einy for his valuable comments and pointing out some related references, and would like to thank an anonymous referee and the editor for their helpful comments. S. Kusuoka, T. 1007/978-4-431-99490-9 2, c Springer 2010 35 36 J. -I. Takekuma proof, neither irreflexivity nor transitivity is assumed on traders’ preference relations. Our purpose is to revisit his equivalence theorem under somewhat different or general assumptions.
Advances in Mathematical Economics by Fettah Akhiat, Charles Castaing, Fatima Ezzaki (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)