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 $1\leq p<+\infty$.

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

Then the following three conditions are equivalent:

1. For all $f\in L_{+}^{p}(\Omega ,{\mathcal {F}},P)$ with $f\neq 0$ exists a constant $c>0$, such that $cf\not \in {\overline {K-B_{+}}}$.
2. For all $A\in {\mathcal {F}}$ with $P(A)>0$ exists a constant $c>0$, such that $cI_{A}\not \in {\overline {K-B_{+}}}$.
3. There exists a random variable $Z\in L^{q}$, such that $Z>0$ almost surely and

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

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.
• Freddy Delbaen and Walter Schachermayer: The Mathematics of Arbitrage (2005). Springer Finance