Unifying framework for optimal insurance

For a given loss X, suppose that one can purchase partial insurance I(X) where 0≤I(x)≤x for all x, subject to a premium principle H(I). The object is to choose I to optimize some quantity G(I,H(I)). A classical problem of this type is a theorem of Arrow that seeks to maximize the expected utility of resulting wealth, when H(I) is some nondecreasing function of E(I). In this paper, we present a unifying framework for determining optimal insurance for general G and H. To perform the required analysis, we consider the notion of the derivative of a functional. This allows us to include previous results within our framework, including Arrow's Theorem, Young's work on Wang's premium principle, and the work of Gajek and Zagrodny on minimizing the variance of retained claims subject to a standard deviation premium principle.