The value at risk (V @ R ) is a very important risk measure with significant applications in finance (risk management, pricing, hedging, portfolio theory, etc), insurance (premium principles, optimal reinsurance, etc), production, marketing (newsvendor problem), etc. It also plays a critical role in regulation about risk (Basel, Solvency, etc), it is very appreciated by practitioners due to its intuitive interpretation, and it is the unique popular risk measure remaining finite for heavy tailed risks with unbounded expectation. Besides, ambiguous frameworks are becoming more and more usual in applications of risk analysis. Lack of data or committed errors may provoke discrepancies between real probabilities and estimated ones. This paper combines both V @ R and ambiguous settings, and a new representation theorem for V @ R is given. Consequently, inspired by previous studies dealing with coherent risk measures and their representation, we will give new methods to compute and optimize V @ R under ambiguity. This seems to be a relevant finding because the analytical properties of V @ R are very weak if one compares with a coherent risk measure. Indeed, V @ R is neither continuous nor convex, which makes it very complicated to deal with it in mathematical approaches. Nevertheless, the results of this paper will allow us to transform computation and optimization problems involving V @ R into continuous and differentiable problems.
ambiguity; heavy tail; representation theorem; value at risk