Distortion risk measures of step-weighted distribution

Battle Count: 84

The paper provides theoretical foundations for computing VaR and TVaR of step-weighted distributions and their comonotonic sums, which are relevant to portfolio risk aggregation and tail risk assessment. However, it is purely theoretical with no direct trading strategy, backtesting, or market data analysis. The results on comonotonic sums of lognormal variables could inform risk models for derivative portfolios, but practical implementation guidance is absent.

The main results (Theorems 1.2, 1.3, 1.5) provide explicit formulas for computing distortion risk measures of step-weighted distributions. The construction of G_{A,Q} and g_{A,Q} is straightforward given vectors A and Q. Corollary 4.6 gives closed-form VaR and TVaR expressions. However, the paper provides no code, and implementing the general distortion risk measure framework requires careful handling of Lebesgue-Stieltjes integrals and generalized inverses.

The paper is purely theoretical with complete mathematical proofs. All theorems, lemmas, and corollaries are fully proved. Explicit formulas for VaR and TVaR of step-weighted distributions are provided. Numerical examples (Example 4.7, 5.4, 5.5) allow verification. However, no code or computational tools are provided.