Before revision:  We develop results that guarantee the existence of a solution , indicate properties that our risk measure fulfills, and characterize the resulting minimum cost as a deviation measure. We generalize this approach to a robust framework, where the minimization is over a supremum of expectations, based on a convex set of probability measures .
After revision:  We develop results that guarantee the existence of a solution and explore the properties of minimizer and minimum as risk and deviation measures, respectively .