Further Applications

Abstract

In this chapter, we show that coherent risk measures in mathematical finance can be formulated using translation invariant functionals such that it is possible to use the results proved in this book in order to derive corresponding properties for coherent risk measures. Furthermore, we study the relationship between coherent risk measures and a strictly robust counterpart problem of an optimization problem under uncertainty. The benefit function and shortage function in mathematical economics are related to translation invariant functionals. Moreover, we consider a vector-valued optimal control problem with PDE-constraints and apply the scalarization technique by means of translation invariant functionals for deriving characterizations of solutions to this vector-valued optimal control problem that are useful for corresponding adaptive algorithms. Finally, we use the directional minimal time function for the formulation of location problems and present necessary optimality conditions for solutions of these (in general nonconvex) location problems.