In this dissertation we establish some links between the non-additive integration theory and some useful notions in finance and insurance, such as the notions of stochastic ordering and risk measure. In the framework of ambiguity, the notion of capacity (or non-additive probability) replaces that of probability measure, and Choquet integrals replace the usual mathematical expectations. In this thesis, we extend the notions of increasing, and increasing convex stochastic dominance, well-known in the case of a probability, to this more general framework. We characterize these relations in terms of distribution functions and quantile functions with respect to the initial capacity. We also establish a generalization of Hardy-Littlewood's inequalities to the case of a capacity, which we apply in solving an optimization problem whose constrains are given by means of the "generalized" increasing convex relation. We are then interested in the classes of monetary risk measures having the properties of comonotonic additivity and consistency with respect to a given "generalized" stochastic dominance relation. These are characterized in terms of Choquet integrals with respect to a distorted capacity. A Kusuoka-type characterization of the class of risk measures having the properties of comonotonic additivity and consistency with respect to the "generalized" increasing convex ordering is also established. Finally, we are interested in those risk measures that have a "robust" representation as a maximum, over a set of distortion functions, of Choquet integrals with respect to a distortion of the initial capacity.
