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.

Yan's theorem

Notation:

Statement

Let be a probability space, and be the space of non-negative and bounded random variables. Further let be a convex subset and.
Then the following three conditions are equivalent:
  1. For all with exists a constant, such that.
  2. For all with exists a constant, such that.
  3. There exists a random variable, such that almost surely and

Literature