By Shigeo Kusuoka, Toru Maruyama
The sequence is designed to collect these mathematicians who're heavily drawn to getting new not easy stimuli from monetary theories with these economists who're looking powerful mathematical instruments for his or her study. loads of financial difficulties may be formulated as limited optimizations and equilibration in their suggestions. numerous mathematical theories were providing economists with necessary machineries for those difficulties coming up in financial idea. Conversely, mathematicians were encouraged by way of quite a few mathematical problems raised by way of monetary theories
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Extra resources for Advances in mathematical economics. Vol.19
Acknowledgements We are grateful to Toru Maruyama for his helpful comments and suggestions. We would also like to express gratitude to the anonymous referee for their helpful advice. A Theory for Estimating Consumer’s Preference from Demand 55 References 1. Afriat S (1967) The construction of a utility function from demand data. Int Econ Rev 8:67–77 2. Antonelli GB (1886) Sulla Teoria Matematica Dell’ Economia Politica. Tipografia del Folchetto, Pisa 3. Debreu G (1972) Smooth preferences. Econometrica 40:603–615 4.
Step 3: choose any compact subset V of 2 . Fix any " > 0. x; v/k < kvk"g, where y1 ; y2 are as in Lemma 2. Because y1 ; y2 is continuous, W is open in V , and thus V 0 D V n W is compact. x; v/I x; v/k kvk Ä ": Ä 17 See Sect. 3. x; v/ 2 V 0 . x; v/ 2 W , a contradiction. Hence, x is not proportional to v. x; v/ is continuous on V 0 . x; v/ in V 0 such that y. x;v/ . x;v/ , yk . x; v/, and (iii) For any k k0 , Eq. x; v/. xi ;vi / D V 0 . x; v/j Ä ". xi ; vi / as ci . x; v/ is in the domain of yk if k is sufficiently large.
N n 1X i 1X i X˛k ; EŒX˛1k / C max dH . X C ; EŒX 1C / ˛k 1 1ÄkÄm n i D1 n i D1 ˛k 1 n Ä max dH . EŒX 1C ; EŒX˛1k / WD I1 C I2 C I3 : 1ÄkÄm ˛k 1 Now the rest of the proof is identical to the last part of the proof of Theorem 15 by noting that I3 is Ä 2" and I1 ! 0 and I2 ! 0 when n ! 1. 6 Fuzzy Martingale and Integrand Martingale We discuss in this section the concept of fuzzy martingale and integrand martingale and provide some related convergence results. Fn /n2N be an increasing sequence of sub -algebras of F such that F is the -algebra generated by [n2N Fn .