Bidual Approaches in Risk Representation

Downside and deviation risk measures are becoming more and more important in many disciplines with clear interfaces with Applied Mathematics and Operations Research. Their dual representations have played critical roles in most of their applications (risk management, portfolio selection, pricing and hedging, etc.), but, to the best of our knowledge, bidual representations were never profoundly studied. New linear bidual representations will be provided, and their great capacity to linearize many problems will be proved, with special focus on risk optimization. This is important because there are very tractable necessary and sufficient optimality conditions and resolution algorithms in Linear Programming. Moreover, in the linearization process, one will have to introduce new decision variables providing us with very important information, such as sensitivities with respect to the selected risk measure and sensitivities with respect to the selected model (model risk). The theory will be presented for general Banach spaces, and an illustrative example will be given.