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Yan's theorem

From Wikipedia, the free encyclopedia

In probability theory, Yan's theorem is a separation and existence result. It is of particular interest in financial mathematics where one uses it to prove the Kreps-Yan theorem.

The theorem was published by Jia-An Yan.[1] It was proven for the L1 space and later generalized by Jean-Pascal Ansel to the case .[2]

Yan's theorem

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Notation:

is the closure of a set .
.
is the indicator function of .
is the conjugate index of .

Statement

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Let be a probability space, and be the space of non-negative and bounded random variables. Further let be a convex subset and .

Then the following three conditions are equivalent:

  1. For all with exists a constant , such that .
  2. For all with exists a constant , such that .
  3. There exists a random variable , such that almost surely and
.

Literature

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  • Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de ou ". Séminaire de probabilités de Strasbourg. 14: 220–222.
  • Freddy Delbaen and Walter Schachermayer: The Mathematics of Arbitrage (2005). Springer Finance

References

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  1. ^ Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de ou ". Séminaire de probabilités de Strasbourg. 14: 220–222.
  2. ^ Ansel, Jean-Pascal; Stricker, Christophe (1990). "Quelques remarques sur un théorème de Yan". Séminaire de Probabilités XXIV, Lect. Notes Math. Springer.