Continuity of VaR and Continuous Differentiability of CVaR under Decision-Dependent Losses

2026-06-08

AI Summary

This work addresses the challenge that when the loss function depends on decision variables, the regularity properties—such as continuity and differentiability—of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are generally not guaranteed, thereby hindering theoretical and algorithmic advances in risk-aware optimization. Focusing on such decision-dependent losses, the paper establishes simple yet rigorous sufficient conditions under which CVaR is proven for the first time to be continuously differentiable, and provides an explicit expression for its gradient. By integrating tools from perturbation analysis of probability measures, path-differentiability, and real analysis, the study develops a unified theoretical framework that ensures the continuity of VaR and the continuous differentiability of CVaR, thereby furnishing reliable gradient information and convergence guarantees for optimization problems involving tail risk.

Abstract

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are widely used in risk-aware optimization and equilibrium models. When the loss depends on a decision variable, the induced distribution, the VaR threshold, and the CVaR tail set all change with the decision. This makes the regularity of the VaR and CVaR maps nontrivial. We give simple sufficient conditions under which the VaR map is continuous and the corresponding CVaR map is continuously differentiable. The assumptions are local around the VaR level and rely on dominated pathwise differentiability of the scenario-wise loss. We also derive the CVaR gradient formula, thereby justifying first-order analysis for decision-dependent tail-risk models.

Problem

• VaR
• CVaR
• decision-dependent losses
• continuity
• continuous differentiability

Innovation

• decision-dependent losses
• Value-at-Risk (VaR)
• Conditional Value-at-Risk (CVaR)
• continuous differentiability
• risk-aware optimization