Weak Continuity of Risk Functionals with Applications to Stochastic Programming

Claus, Matthias; Krätschmer, Volker; Schultz, Rüdiger

doi:10.1137/15m1048689

Cited by 13 publications

(6 citation statements)

References 42 publications

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“…In the present work, we shall follow the approach of [7] and confine the stability analysis to locally uniformly • p -integrating sets. A detailed discussion of locally uniformly • p -integrating sets is provided in [15], [26], [27], and [28].…”

Section: A Stability Results For Bilevel Stochastic Linear Problemsmentioning

confidence: 99%

Risk-Averse Models in Bilevel Stochastic Linear Programming

Burtscheidt¹,

Claus²,

Dempe³

2020

SIAM J. Optim.

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We consider bilevel linear problems, where some parameters are stochastic, and the leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader's outcome can be modeled by a random variable, which we evaluate based on some lawinvariant convex risk measure. A qualitative stability result under perturbations of the underlying probability distribution is presented. Moreover, for the expectation, the expected excess, and the upper semideviation, we establish Lipschitz continuity as well as sufficient conditions for differentiability. Finally, for finite discrete distributions, we reformulate the bilevel stochastic problems as standard bilevel problems and propose a regularization scheme for bilevel linear problems.

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Section: A Stability Results For Bilevel Stochastic Linear Problemsmentioning

confidence: 99%

Risk-Averse Models in Bilevel Stochastic Linear Programming

Burtscheidt¹,

Claus²,

Dempe³

2020

SIAM J. Optim.

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“…5, we address stochastic programming problems. It can be inferred from results of Claus et al [9] that the optimal value of a general stochastic programming problem depends copula robustly on the distribution of the underlying d-variate input random variable Z. This covers in particular classical one-period portfolio optimisation problems (where the role of Z is played by the vector of the relative price changes of d risky assets) and therefore backs in a way a hypothesis of Saida and Prigent [45].…”

Section: F μ D (X D )mentioning

confidence: 99%

A concept of copula robustness and its applications in quantitative risk management

Zähle

2022

Finance Stoch

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In financial and actuarial applications, marginal risks and their dependence structure are often modelled separately. While it is sometimes reasonable to assume that the marginal distributions are ‘known’, it is usually quite involved to obtain information on the copula (dependence structure). Therefore copula models used in practice are quite often only rough guesses. For many purposes, it is thus relevant to know whether certain characteristics derived from $d$ d -variate risks are robust with respect to (at least small) deviations in the copula. In this article, a general concept of copula robustness is introduced and criteria for copula robustness are presented. These criteria are illustrated by means of several examples from quantitative risk management. The concept of aggregation robustness introduced by Embrechts et al. (Finance Stoch. 19:763–790, 2015) can be embedded in our framework of copula robustness.

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“…As (4) is strictly feasible for any right-hand side t ∈ R s , strong duality holds and (5) implies that the infimum of (4) is zero. Furthermore, for any t ∈ R \ {0} we have…”

Section: Remarkmentioning

confidence: 99%

“…holds for any bounded and continuous function h : R s → R. It is well known that even for linear recourse one cannot expect weak continuity of Q R on the entire space S n + × M p s . Along the lines of [5], we shall thus restrict the analysis to appropriate subspaces.…”

Section: ⊓ ⊔mentioning

confidence: 99%

On Risk-Averse Stochastic Semidefinite Programs with Continuous Recourse

et al. 2019

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The vast majority of the literature on stochastic semidefinite programs (stochastic SDPs) with recourse is concerned with risk-neutral models. In this paper, we introduce mean-risk models for stochastic SDPs and study structural properties as convexity and (Lipschitz) continuity. Special emphasis is placed on stability with respect to changes of the underlying probability distribution. Perturbations of the true distribution may arise from incomplete information or working with (finite discrete) approximations for the sake of computational efficiency. We discuss extended formulations for stochastic SDPs under finite discrete distributions, which turn out to be deterministic (mixed-integer) SDPs that are (almost) block-structured for many popular risk measures.

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Weak Continuity of Risk Functionals with Applications to Stochastic Programming

Cited by 13 publications

References 42 publications

Risk-Averse Models in Bilevel Stochastic Linear Programming

Risk-Averse Models in Bilevel Stochastic Linear Programming

A concept of copula robustness and its applications in quantitative risk management

On Risk-Averse Stochastic Semidefinite Programs with Continuous Recourse

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