  


Frechet inequalities via convex optimization
Bart Paul Gerard Van Parys(vanparysmit.edu) Abstract: Quantifying the risk carried by an aggregate position $S_d\defn\sum_{i=1}^d X_i$ comprising many risk factors $X_i$ is fundamental to both insurance and financial risk management. Frechet inequalities quantify the worstcase risk carried by the aggregate position given distributional information concerning its composing factors but without assuming independence. This marginal factor modeling of the aggregate position in terms of its risk factors $X_i$ leaves, however, the distribution of $S_d$ ambiguous. The resulting distributional ambiguity can be traced back directly to the lack of dependence structure imposed between the risk factors. Frechet inequalities implicitly uphold already the work of George Boole on probabilistic logic but were explicitly derived by Maurice Frechet only in 1935. These inequalities can be considered rules about how to bound calculations involving probabilities without assuming independence or, indeed, without making any dependence assumptions whatsoever. It is exactly this robustness to obscure risk dependencies which have stimulated their renewed interest in the post crisis economy. We will approach these Frechet inequalities using a modern convex optimization lens. Our novel perspective provides data friendly computational tools applicable to practical Frechet problems. Two important data friendly classes of Frechet problems are presented, both of which are shown to admit exact tractable convex optimization reformulations. The efficacy of our approach is illustrated using a small insurance management problem. Keywords: Frechet inequalities, convex optimization, distributionally robust optimization Category 1: Robust Optimization Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Category 3: Infinite Dimensional Optimization Citation: Download: [PDF] Entry Submitted: 07/11/2016 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  