137 research outputs found
Uniqueness of Kusuoka Representations
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.
Further, stochastic order relations are employed to identify the minimal
Kusuoka representation. It is shown that measures in the minimal representation
are extremal with respect to the order relations. The tools are finally
employed to provide the minimal representation for important practical
examples. Although the Kusuoka representation is usually given only for
nonatomic probability spaces, this presentation closes the gap to spaces with
atoms
Optimization with multivariate conditional value-at-risk constraints
For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice.
As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to
finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the
proposed solution methods
Optimization with multivariate conditional value-at-risk constraints
For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice.
As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to
finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the
proposed solution methods
Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems
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
Comonotonic Measures of Multivariate Risks.
We propose a multivariate extension of a well-known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invari- ance, subadditivity and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we refor- mulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.Comonotonicity; Maximal Correlation; Optimal Transportation; Regular Risk Measures; Coherent Risk Measures;
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