MASTROGIACOMO ELISA

Decision making is ubiquitous in daily life as well as economics, finance, social and natural sciences. Finding best alternatives in a set which also includes non-comparable elements, i.e., w.r.t. an incomplete preference, is especially complicated. Incomplete preferences and multi-criteria decision making problems were studied for quite some time. Recent developments motivate to address them via new approaches based on set-valued functions. My research comprises three major topics. - Optimization of vector/set-valued functions. By set-valued analysis and vector/set optimization techniques, optimality conditions as well as duality theorems for optimization of convex set-valued risk functions shall be established which requires, e.g., the development of new set-valued derivatives and advanced scalarization techniques. Moreover, it shall be investigated how solutions of vector/set-valued optimization problems react to perturbations in the data (robustness). Applications include finding risk minimal portfolios in market models with frictions, capital allocation via set-valued risk measures as well as the dependence of such problems on the underlying probability measure. - Set-valued multivariate statistics and elicitability of set-valued risk measures. Based on a new approach to multivariate statistics, results for recently introduced lower cone distribution functions and set-valued quantiles as well as links to stochastic orders and set-valued risk measures shall be established. Moreover, efficient numerical procedures based on computational geometry methods shall be provided. Estimation, asymptotic analysis and elicitability of set-valued risk measures/quantiles will be studied with tools from robust statistics and probability theory for random sets (extended if needed). Tools from the previous topic will be essential for this. - Dynamic optimization problems. Based on the maximum principle and variational techniques, a new general theory of time-inconsistent controlled FBSDEs under general state constraints shall be developed with applications to financial decision making. A generalized notion of optimality in terms of subgame-perfect equilibrium shall be required. Capital allocation for general risk measures shall be studied in a dynamic setting and defined by means of generalized gradients introduced axiomatically. For those induced by BSDEs and BSVIEs, corresponding dynamic rules shall be derived and characterized in terms of the driver. Generalizations to a set-valued framework will follow.