By S. Kusuoka, A. Yamazaki

ISBN-10: 4431343415

ISBN-13: 9784431343417

ISBN-10: 4431343423

ISBN-13: 9784431343424

Loads of monetary difficulties can formulated as restricted optimizations and equilibration in their options. quite a few mathematical theories were delivering economists with integral machineries for those difficulties coming up in monetary thought. Conversely, mathematicians were encouraged by way of a variety of mathematical problems raised through monetary theories. The sequence is designed to collect these mathematicians who have been heavily attracted to getting new not easy stimuli from monetary theories with these economists who're looking for powerful mathematical instruments for his or her researchers.

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**Additional info for Advances in Mathematical Economics, Volume 9 (Advances in Mathematical Economics)**

**Example text**

53 (3) The functions ^7^, 0 < a < 1, introduced in (2) provide the simplest example of law invariant monetary utility functions, which correspond to the so-called average value at risk. 2 Strong and weak upper semi-continuity of law invariant maps We now have assembled all the concepts that are needed to formulate our first main result. 1. Suppose that (^2,^, P) is a standard probability space. For a function U: L°°(f], ^ , P ) -^ R the following are equivalent: (i) U is a law invariant monetary utility function.

C. with respect to the topology induced by \\. c. with respect to the a{L'^,h^)'topology. We prove this result, which we consider as the main contribution of this paper, in section 4. 2 may directly proceed to this section. 1 implies in particular that in Theorem 7 of [KOI] the assumption of the Fatou property may also be dropped. For the sake of completeness we formulate this result. 54 E. Jouini, W. Schachermayer, N. Touzi A monetary utility U{Xi + X2) = U{Xi) pare [JST05]). , U{XX) = a coherent risk measure.

Fc^o - ( • /or any p G (1, oo) and n € N. )}] by Jensen's inequality. =Y. (£),j/,)}]|^,^y. =y. =Yy Pi = 0. k=0 Therefore we have our assertion. This completes the proof. D 38 H. Fushiya Now we prove Theorem 1. Let h^^\y) = Y^h^i\y)h^2 ^\yX where j=0 hi, /i2 are as in Propositions 3 and 4. Then we have limE ^{E[g{X,{e)) £->0 - I Gtis)] -j^e'h^'Hne))] k=o ^ -0, ^ for any p G (1, oo) and n G N. At last we show the uniqueness of h^^^. Suppose that h^^\k = 0^ 1, 2 , . . , also satisfy = 0, limE £-^0 for any p € (1, oo) and n € N .

### Advances in Mathematical Economics, Volume 9 (Advances in Mathematical Economics) by S. Kusuoka, A. Yamazaki

by Brian

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