Paper: arXiv 2511.00717

Authors: Peng Liu, Alexander Schied

Abstract

In this paper, we investigate the Lambda Value-at-Risk ($Λ$VaR) under ambiguity, where the ambiguity is represented by a family of probability measures. We establish that for increasing Lambda functions, the robust (i.e., worst-case) $Λ$VaR under such an ambiguity set is equivalent to $Λ$VaR computed with respect to a capacity, a novel extension in the literature. This framework unifies and extends both traditional $Λ$VaR and Choquet quantiles (Value-at-Risk under ambiguity). We analyze the fundamental properties of this extended risk measure and establish a novel equivalent representation for $Λ$VaR under capacities with monotone Lambda functions in terms of families of downsets. Moreover, explicit formulas are derived for robust $Λ$VaR when ambiguity sets are characterized by $φ$-divergence and the likelihood ratio constraints, respectively. We further explore the applications in risk sharing among multiple agents. We demonstrate that the family of risk measures induced by families of downsets is closed under inf-convolution. In particular, we prove that the inf-convolution of $Λ$VaR with capacities and monotone Lambda functions is another$Λ$VaR under a capacity. The explicit forms of optimal allocations are also derived. Moreover, we obtain more explicit results for risk sharing under ambiguity sets characterized by $φ$-divergence and likelihood ratio constraints. Finally, we explore comonotonic risk-sharing for $Λ$VaR under ambiguity.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is a rigorous mathematical investigation of Lambda Value-at-Risk under model ambiguity and capacity theory, establishing fundamental analytical properties and explicit inf-convolution formulas. However, it is entirely theoretical with no empirical data analysis, simulation, or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Robust Lambda Value-at-Risk (ΛVaR) under Ambiguity & Risk Sharing"] --> B["Inputs: Monotone Λ Functions & Ambiguity Sets (φ-Divergence, Likelihood Ratio)"]
    B --> C["Theoretical Modeling: Capacity Equivalence & Downset Families Representation"]
    C --> D["Inf-Convolution & Risk Sharing Optimization"]
    D --> E["Key Findings: Explicit Robust ΛVaR Formulas & Closed-Form Optimal Risk Allocations"]