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.^[1] It was proven for the L¹ space and later generalized by Jean-Pascal Ansel to the case ${\displaystyle 1\leq p<+\infty }${\displaystyle 1\leq p<+\infty }.^[2]

Notation:

${\displaystyle {\overline {\Omega }}}${\displaystyle {\overline {\Omega }}} is the closure of a set ${\displaystyle \Omega }${\displaystyle \Omega }.

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

${\displaystyle I_{A}}${\displaystyle I_{A}} is the indicator function of ${\displaystyle A}${\displaystyle A}.

${\displaystyle q}${\displaystyle q} is the conjugate index of ${\displaystyle p}${\displaystyle p}.

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

Then the following three conditions are equivalent:

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

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

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