Inf-convolution and Optimal Allocations for Tail Risk Measures

Liu, Fangda; Mao, Tiantian; Wang, Ruodu; Wei, Linxiao

doi:10.2139/ssrn.3490348

Cited by 5 publications

(8 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For p > 1, Since the Wasserstein ball is convex, it follows from Theorem 1 that sup F ∈Fp,ε(F 0 ) ES α (F ) = ES α (F 2 p,ε|F 0 ). By Proposition 4 (ii) of Liu et al (2022), we have sup F ∈Fp,ε(F 0 ) ES α (F ) = ES α (F 0 ) + (1 − α) −1/p ε for α ∈ (0, 1). Therefore, one can obtain For any F ∈ F w,a,p,ε (F X ), by definition, there exists Z with F Z ∈ F d a,p,ε (F X ) and F = F w Z .…”

Section: Suppose By Contradiction Thatmentioning

confidence: 97%

“…The risk measure ρ is defined by ρ h (F ) = 1 0 VaR s (F )dh(s), where h : [0, 1] → [0, 1] is increasing and convex with h(0) = 1 − h(1) = 0. Noting that ρ h is translation invariant and positively homogeneous, the WR portfolio optimization problem is, by applying Proposition 4 of Liu et al (2022) and Theorem 7,…”

Section: Multivariate Wasserstein Uncertaintymentioning

confidence: 99%

“…Since the distribution functions F 1 p,ε|F 0 and F 2 p,ε|F 0 , as well as their quantile functions, are obtained explicitly in Theorem 6, the robust risk values ρ MA 1 (F p,ε (F 0 )) and ρ MA 2 (F p,ε (F 0 )) can be computed for any risk measure ρ in a straightforward manner. On the other hand, ρ WR (F p,ε (F 0 )) is often difficult to compute if the risk measure is complicated, although there are some results in the literature considered the WR approach for special choices of risk measures; see Liu et al (2022) for distortion risk measures, and Gao and Kleywegt (2016), Esfahani and Kuhn (2018) and for robust stochastic optimization.…”

Section: Uncertainty Induced By the Univariate Wasserstein Metricmentioning

confidence: 99%

See 2 more Smart Citations

Model Aggregation for Risk Evaluation and Robust Optimization

Mao¹,

Wang²,

Wu³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Suppose By Contradiction Thatmentioning

confidence: 97%

Section: Multivariate Wasserstein Uncertaintymentioning

confidence: 99%

Section: Uncertainty Induced By the Univariate Wasserstein Metricmentioning

confidence: 99%

See 1 more Smart Citation

Model Aggregation for Risk Evaluation and Robust Optimization

Mao¹,

Wang²,

Wu³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Š. Schwarz in [7] have introduced the convolution measures to classify the available boundary between the disordered and ordered stages (in randomized Boolean network), which integrates variance. This strategy might provide a remedy to the calls for interconnectivity measures meant to integrate the matrix implications (spatiotemporary heterogeneity) on landscale and Metapopulace convolution.…”

Section: Fig 1 the Convolution Profile Of Systems With Sosmentioning

confidence: 99%

Complex Dynamic Systems Theory for Cognitive Environment Approach

2021

JCNS

View full text Add to dashboard Cite

Population Fluctuations (PF), Patch Variation (PV), and Food Webs (FW) are just a few of the areas where the Complex Dynamic Systems Theory (CDST) has made a significant impact on our understanding of the environment. Measures have been used to capture the variation between simple, disordered and ordered frameworks with local interactions that can generate surprising actions on a massive scale. But research shows that conventional explanations of convolution fail to take into account some major characteristics of ecological systems, an ideology that will limit the contributions of CDST to the entire ecosystem. In this paper, we have presented literature review of these characteristics of Environmental Convolution (EC), e.g. diversification, environmental variability, memory and cross-scale interactions, which progress to classical CDST. Advancements in these segments will be essential before CDST can be applicable in the comprehension of more vibrant systems in the environment.

show abstract

“…Beyond the usual approach of convex risk measures, some studies have been concerned with inf-convolution in relation to specific properties, as in (Acciaio, 2007), (Grechuk et al, 2009), (Grechuk and Zabarankin, 2012), (Carlier et al, 2012), (Mastrogiacomo and Rosazza Gianin, 2015), and (Liu et al, 2020), particular risk measures, as the recent quantile risk sharing in (Embrechts et al, 2018b), (Embrechts et al, 2018a), (Weber, 2018), (Wang and Ziegel, 2018), and (Liu et al, 2019), or even specific topics, as in (Wang, 2016) and (Liebrich and Svindland, 2019). However, these studies, with or without convexity, are restricted to finite (or at most countable in rare cases) sets of risk measures.…”

Section: Introductionmentioning

confidence: 99%

Inf-convolution and optimal risk sharing with countable sets of risk measures

Righi,

Moresco

2020

Preprint

View full text Add to dashboard Cite

The inf-convolution of risk measures is directly related to risk sharing and general equilibrium, and it has attracted considerable attention in mathematical finance and insurance problems. However, the theory is restricted to finite (or at most countable in rare cases) sets of risk measures. In this study, we extend the inf-convolution of risk measures in its convex-combination form to an arbitrary (not necessarily finite or even countable) set of alternatives. The intuitive principle of this approach is to regard a probability measure as a generalization of convex weights in the finite case. Subsequently, we extensively generalize known properties and results to this framework. Specifically, we investigate the preservation of properties, dual representations, optimal allocations, and self-convolution.

show abstract

Inf-convolution and Optimal Allocations for Tail Risk Measures

Cited by 5 publications

References 36 publications

Model Aggregation for Risk Evaluation and Robust Optimization

Model Aggregation for Risk Evaluation and Robust Optimization

Complex Dynamic Systems Theory for Cognitive Environment Approach

Inf-convolution and optimal risk sharing with countable sets of risk measures

Contact Info

Product

Resources

About