Conditional Distributionally Robust Functionals

Risk measures incorporate a conservative or risk averse perspective in decisionmaking under uncertainty. Taking a variety of models for the potential outcomes into account, the distributionally robust decision is the most conservative decision among the decisions available. This paper investigates different versions of conditional risk measures and distributionally robust functionals in a multistage setting. The … Read more

Risk-Averse Optimal Control

In the context of multistage stochastic optimization, it is natural to consider nested risk measures, which originate by repeatedly composing risk measures, conditioned on realized observations. Starting from this discrete time setting, we extend the notion of nested risk measures to continuous time by adapting the risk levels in a time dependent manner. This time … Read more

Risk averse stochastic programming: time consistency and optimal stopping

Bellman formulated a vague principle for optimization over time, which characterizes optimal policies by stating that a decision maker should not regret previous decisions retrospectively. This paper addresses time consistency in stochastic optimization. The problem is stated in generality first. The paper discusses time consistent decision-making by addressing risk measures which are recursive, nested, dynamically … Read more

Dynamic Scaling and Submodel Selection in Bundle Methods for Convex Optimization

Bundle methods determine the next candidate point as the minimizer of a cutting model augmented with a proximal term. We propose a dynamic approach for choosing a quadratic proximal term based on subgradient information from past evaluations. For the special case of convex quadratic functions, conditions are studied under which this actually reproduces the Hessian. … Read more

An Analytical Study of Norms and Banach Spaces Induced by the Entropic Value-at-Risk

This paper addresses the Entropic Value-at-Risk (EVaR), a recently introduced coherent risk measure. It is demonstrated that the norms induced by EVaR induce the same Banach spaces, irrespective of the confidence level. Three spaces, called the primal, dual, and bidual entropic spaces, corresponding with EVaR are fully studied. It is shown that these spaces equipped … Read more

Quantitative Stability Analysis for Minimax Distributionally Robust RiskOptimization

This paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage. We carry out stability analysis by looking into variations of the ambiguity set under the Wasserstein metric, decision spaces at both stages and the support … Read more

Time and Dynamic Consistency of Risk Averse Stochastic Programs

In various settings time consistency in dynamic programming has been addressed by many authors going all the way back to original developments by Richard Bellman. The basic idea of the involved dynamic principle is that a policy designed at the first stage, before observing realizations of the random data, should not be changed at the … Read more

A Quantitative Comparison of Risk Measures

The choice of a risk measure reflects a subjective preference of the decision maker in many managerial, or real world economic problem formulations. To evaluate the impact of personal preferences it is thus of interest to have comparisons with other risk measures at hand. This paper develops a framework for comparing different risk measures. We … Read more

Dynamic Generation of Scenario Trees

We present new algorithms for the dynamic generation of scenario trees for multistage stochastic optimization. The different methods described are based on random vectors, which are drawn from conditional distributions given the past and on sample trajectories. The structure of the tree is not determined beforehand, but dynamically adapted to meet a distance criterion, which … Read more

Minimal Representation of Insurance Prices

This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness — in a sense specified in the paper — of the risk measure’s Kusuoka representation is derived from this initial result. … Read more