Yan's theorem

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. It was proven for the L¹ space and later generalized by Jean-Pascal Ansel to the case $1\le p<+\mathrm{\infty }$.

Yan's theorem

Notation:

$\bar{\mathrm{\Omega }}$ is the closure of a set $\mathrm{\Omega }$.

$A-B=\{f-g:f\in A,g\in B\}$.

$I_A$ is the indicator function of $A$.

$q$ is the conjugate index of $p$.

Statement

Let $(\mathrm{\Omega },{ℱ},P)$ be a probability space, $1\le p<+\mathrm{\infty }$ and $B_{+}$ be the space of non-negative and bounded random variables. Further let $K\subseteq L^p(\mathrm{\Omega },{ℱ},P)$ be a convex subset and $0\in K$.

Then the following three conditions are equivalent:

1. For all $f\in L_{+}^p(\mathrm{\Omega },{ℱ},P)$ with $f\ne 0$ exists a constant $c>0$, such that $cf\in ̸\overline{K-B_{+}}$.
2. For all $A\in {ℱ}$ with $P(A)>0$ exists a constant $c>0$, such that $c{I}_A\in ̸\overline{K-B_{+}}$.
3. There exists a random variable $Z\in L^q$, such that $Z>0$ almost surely and

$\sup {}_{Y\in K}\mathbb{𝔼}[ZY]<+\mathrm{\infty }$.

Literature

• Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de $L^1$ ou $H^1$". Séminaire de probabilités de Strasbourg 14: 220–222. http://www.numdam.org/item/SPS_1980__14__220_0/. 
• Freddy Delbaen and Walter Schachermayer: The Mathematics of Arbitrage (2005). Springer Finance