Minimizing vector risk measures

Abstract

The minimization of risk functions is becoming very important due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Vector optimization problems involving many types of risk functions are studied. The "balance space approach" of multiobjective optimization and a general representation theorem of risk functions is used in order to transform the initial minimization problem in an equivalent one that is convex and usually linear. This new problem permits us to characterize optimality by saddle point properties that easily apply in practice. Applications in finance and insurance are presented. © Springer-Verlag Berlin Heidelberg 2010.