Yan's theorem explained

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

1 \leq p< +inf ty .^[2]

Yan's theorem

Notation:

\overline{\Omega} is the closure of a set \Omega .A - B = \{f - g:f\in A, g\in B\} .I _{A} is the indicator function of A .q is the conjugate index of p .

Statement

Let

(\Omega, l{F},P) be a probability space, 1 \leq p< +inf ty and B _{+} be the space of non-negative and bounded random variables. Further let K\subseteq L ^{p(\Omega,l{F},P)} be a convex subset and 0 \in K .

Then the following three conditions are equivalent:

1. For all

f\in L ^{p(\Omega, l{F},P)} _{+} with f ≠ 0 exists a constant c> 0 , such that cf \not\in \overline{K - B _{+}} .

1. For all

A\in l{F} with P(A)> 0 exists a constant c> 0 , such that cI _{A} \not\in \overline{K - B _{+}} .

1. There exists a random variable

Z\in L ^{q} , such that Z> 0 almost surely and \sup\limits _{Y\in} E [ZY]< +inf ty .

Literature

• Jia-An. Yan. 1980. Caracterisation d' une Classe d'Ensembles Convexes de L ^{1} ou H ^{1} . 14. Séminaire de probabilités de Strasbourg. 220–222.
• Freddy Delbaen and Walter Schachermayer: The Mathematics of Arbitrage (2005). Springer Finance

References

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