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.

翻译：风险价值（VaR）和条件风险价值（CVaR）广泛应用于风险感知优化与均衡模型中。当损失依赖于决策变量时，诱导分布、VaR阈值及CVaR尾部集合均随决策变化而改变，这使得VaR与CVaR映射的正则性变得非平凡。本文给出使得VaR映射连续且对应CVaR映射连续可微的简单充分条件。该假设以VaR水平为中心局部成立，且依赖于场景损失的控制路径可微性。本文同时推导了CVaR梯度公式，从而为决策依赖尾部风险模型的一阶分析提供了理论依据。