Model Aggregation for Risk Evaluation and Robust Optimization

Abstract

We introduce a new approach for prudent risk evaluation based on stochastic dominance, which will be called the model aggregation (MA) approach. In contrast to the classic worstcase risk (WR) approach, the MA approach produces not only a robust value of risk evaluation but also a robust distributional model which is useful for modeling, analysis and simulation, independent of any specific risk measure. The MA approach is easy to implement even if the uncertainty set is non-convex or the risk measure is computationally complicated, and it provides great tractability in distributionally robust optimization. Via an equivalence property between the MA and the WR approaches, new axiomatic characterizations are obtained for a few classes of popular risk measures. In particular, the Expected Shortfall (ES, also known as CVaR) is the unique risk measure satisfying the equivalence property for convex uncertainty sets among a very large class. The MA approach for Wasserstein and mean-variance uncertainty sets admits explicit formulas for the obtained robust models, and the new approach is illustrated with various risk measures and examples from portfolio optimization.

Shortfall-Based Wasserstein Distributionally Robust Optimization

“…Indeed, this result implies that this multi-dimensional optimization problem in (5) can also be solved by putting effort into the completely characterized reformulation of (24). Considering the problem with the ambiguity set constructed by the classic Wasserstein metric (the norm defined in the metric is also E P l(ω ξ), we can obtain a similar result by applying Theorem 7 of [10] and Theorem 1 of [15], that is…”

“…where the second equality comes from |e e|= e p e q and e p = 1. Noting that Eu( Z − X p − ε) = Eu(|Y|−ε), from (10) we obtain Eu(λ|Y|−λε) 0 if and only if Eu(|Y|−ε) 0 for any λ > 0. By the arbitrariness of random variable Y and ε 0, we obtain for any random variable X on R Eu(X) 0 ⇔ Eu(λX) 0, ∀λ > 0.…”

“…In particular, to obtain those results, a key result is the projection result of the ambiguity set based on the shortfall-Wasserstein metric. Such projection result for different ambiguity sets have been studied by [10,11]. In this paper, a necessary and sufficient condition for the projection result of the new ambiguity set is given.…”