The Neyman-Pearson lemma for convex expectations

We study the Neyman-Pearson theory for convex expectations (convex risk measures) on $L^{\infty}(\mu)$. Without assuming that the level sets of penalty functions are weakly compact, a new approach different from the convex duality method is proposed to find a representative pair $(Q^{\ast },P^{\ast})$ such that the optimal tests are just the classical Neyman-Pearson tests between the representative probabilities $Q^{\ast}$ and $P^{\ast}$. The key observation is that the feasible test set is compact in the weak$^{\ast}$ topology by a generalized result of Banach-Alaoglu theorem. Then the minimax theorem can be applied and the representative probability $Q^{\ast}$ is found first. Secondly, under the probability $Q^{\ast}$, we find the representative probability measure $P^{\ast}$ by solving a dual problem. Finally, we apply our results to a shortfall risk minimizing problem in an incomplete financial market.