Risk Measures and Robust Optimization Problems

These lecture notes give a survey on recent developments in the theory of risk measures. The first part outlines the general representation theory of risk measures in a static one-period setting. In particular, it provides structure theorems for law-invariant risk measures. Examples include Value at Risk, Average Value at Risk, distortion risk measures, and risk measures arising from robust preferences. The second part analyzes risk measures and associated robust optimization problems in the framework of dynamic financial market models. The concept of efficient hedging, as introduced by Föllmer and Leukert [32] Föllmer , H. ; Leukert , P. Efficient hedging: cost versus shortfall risk . Finance Stoch. 2000 , 4 , 117 – 146 . [CSA] [Crossref] , [Google Scholar], is discussed in terms of the more general framework of convex risk measures. The last two sections are devoted to the construction of optimal investment strategies under Knightian uncertainty.