Extensions of Ohlin's lemma with applications to optimal reinsurance structures (Q1318554)

From the author's introduction and abstract: Let \(X\) and \(Y\) be random variables with cumulative distribution functions \(F\) and \(G\). Ohlin's lemma states that if \(E X= E Y\), and \((F-G)(t)\) has exactly one sign change from \(-\) to \(+\), then \(E u(x)<E u(Y)\) for any convex function \(u\). With \(X= f(Z)\) and \(Y=g(Z)\) denoting the reinsured amounts in respect to some risk \(Z\), this result makes it possible to identify extremal compensation functions \(f\) and \(g\) corresponding to a fixed expected reinsurance compensation. Ohlin's lemma also plays a role in the theory of (stop-loss) ordering of risks. A risk \(Y\) is said to be `more dangerous' than \(X\) if \(E X\leq E Y\) and the distribution functions \(F\) and \(G\) intersect exactly once as stated in Ohlin's lemma. It is a well-known result that if \(Y\) is more dangerous than \(X\), than \(X\) procedes \(Y\) in stop-loss order. We extend Ohlin's lemma in two directions and show how this leads to an extension of known results on optimal reinsurance structures. Among these are the Arrow-Ohlin theorem and the Vajda-Ohlin theorem.