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 [math]\displaystyle{ 1\leq p\lt +\infty }[/math].

Yan's theorem

Notation:

[math]\displaystyle{ \overline{\Omega} }[/math] is the closure of a set [math]\displaystyle{ \Omega }[/math].

[math]\displaystyle{ A-B=\{f-g:f\in A,\;g\in B\} }[/math].

[math]\displaystyle{ I_A }[/math] is the indicator function of [math]\displaystyle{ A }[/math].

[math]\displaystyle{ q }[/math] is the conjugate index of [math]\displaystyle{ p }[/math].

Statement

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

Then the following three conditions are equivalent:

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

[math]\displaystyle{ \sup\limits_{Y\in K}\mathbb{E}[ZY]\lt +\infty }[/math].

Literature

• Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de [math]\displaystyle{ L^1 }[/math] ou [math]\displaystyle{ H^1 }[/math]". 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