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

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