List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Lectures on Stochastic Programming: Modeling and Theory

Currently, stochastic optimization on the one hand and multi-objective optimization on the other hand are rich and well-established special fields of Operations Research. Much less developed, however, is their intersection: the analysis of decision problems involving multiple objectives and stochastically represented uncertainty simultaneously. This is amazing, since in economic and managerial applications, the features of multiple decision criteria and uncertainty are very frequently co-occurring. Part of the existing quantitative approaches to deal with problems of this class apply scalarization techniques in order to reduce a given stochastic multi-objective problem to a stochastic single-objective one. The present article gives an overview over a second strand of the recent literature, namely methods that preserve the multi-objective nature of the problem during the computational analysis. We survey publications assuming a risk-neutral decision maker, but also articles addressing the situation where the decision maker is risk-averse. In the second case, modern risk measures play a prominent role, and generalizations of stochastic orders from the univariate to the multivariate case have recently turned out as a promising methodological tool. Modeling questions as well as issues of computational solution are discussed.

https://homepage.univie.ac.at/walter.gutjahr/papers/SurveyArticle.pdf

Stochastic multi-objective optimization: a survey on non-scalarizing methods

We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit theorems for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest.

Statistical estimation of composite risk functionals and risk optimization problems

We discuss linear regression approaches to the estimation of law-invariant conditional risk measures. Two estimation procedures are considered and compared; one is based on residual analysis of the standard least-squares method, and the other is in the spirit of the M-estimation approach used in robust statistics. In particular, value-at-risk and average value-at-risk measures are discussed in detail. Large sample statistical inference of the estimators is derived. Furthermore, finite sample properties of the proposed estimators are investigated and compared with theoretical derivations in an extensive Monte Carlo study. Empirical results on the real data (different financial asset classes) are also provided to illustrate the performance of the estimators.

/pdf/conditional-value-at-risk-and-average-value-at-risk-2irgtwb1to.pdf

Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics

We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit formulae for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest.

Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems

Law-invariant or version-independent coherent risk or acceptability functionals do not explicitly depend on the underlying probability space and can be considered as functionals of the distribution function. In this paper, we consider estimates of these functionals based on the empirical distribution function and investigate their asymptotic properties.

Nancy Wozabal

Papers

Asymptotic distribution of law-invariant risk functionals

Uniform limit theorems for functions of order statistics