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 ≤ p < + ∞ {\displaystyle 1\leq p<+\infty } .
Yan's theorem Notation:
Ω ¯ {\displaystyle {\overline {\Omega }}} is the closure of a set Ω {\displaystyle \Omega } .
A − B = { f − g : f ∈ A , g ∈ B } {\displaystyle A-B=\{f-g:f\in A,\;g\in B\}} .
I A {\displaystyle I_{A}} is the indicator function of A {\displaystyle A} .
q {\displaystyle q} is the conjugate index of p {\displaystyle p} .
Statement Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space, 1 ≤ p < + ∞ {\displaystyle 1\leq p<+\infty } and B + {\displaystyle B_{+}} be the space of non-negative and bounded random variables. Further let K ⊆ L p ( Ω , F , P ) {\displaystyle K\subseteq L^{p}(\Omega ,{\mathcal {F}},P)} be a convex subset and 0 ∈ K {\displaystyle 0\in K} . Then the following three conditions are equivalent:
For all f ∈ L + p ( Ω , F , P ) {\displaystyle f\in L_{+}^{p}(\Omega ,{\mathcal {F}},P)} with f ≠ 0 {\displaystyle f\neq 0} exists a constant c > 0 {\displaystyle c>0} , such that c f ∉ K − B + ¯ {\displaystyle cf\not \in {\overline {K-B_{+}}}} . For all A ∈ F {\displaystyle A\in {\mathcal {F}}} with P ( A ) > 0 {\displaystyle P(A)>0} exists a constant c > 0 {\displaystyle c>0} , such that c I A ∉ K − B + ¯ {\displaystyle cI_{A}\not \in {\overline {K-B_{+}}}} . There exists a random variable Z ∈ L q {\displaystyle Z\in L^{q}} , such that Z > 0 {\displaystyle Z>0} almost surely and
sup Y ∈ K E [ Z Y ] < + ∞ {\displaystyle \sup \limits _{Y\in K}\mathbb {E} [ZY]<+\infty } .
Literature Yan, Jia-An (1980). "Caracterisation d' une Classe d'Ensembles Convexes de L 1 {\displaystyle L^{1}} ou 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
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