Paper: arXiv 2611.10531

Authors: Chunle Huang

Abstract

In this paper, based on the concept of step-weighted distribution, we introduce a minimum-deformation approach to constructing comonotonic and moment-matching approximations for sums of lognormal random variables. The proposed approach aims to minimize the Pearson deformation of a starting comonotonic approximation subject to moment-matching constraints. We establish several results concerning the existence and uniqueness of minimum-deformation comonotonic approximations and propose a method for determining the step quantiles and weights. In particular, when the number of steps is one greater than the moment-matching order, we show that the joint optimization problem can be reduced to an optimization problem involving only the partition points. The proposed framework accommodates moment matching of any prescribed finite order, and numerical experiments demonstrate its application to approximations matching the first five moments of the original sum. The numerical results also show that minimizing the Pearson deformation does not necessarily improve the approximation accuracy of risk measures, which depends on both the starting comonotonic approximation and the choice of partition points and correction weights.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical approach to approximating sums of lognormal variables, focusing on theoretical derivations for existence and uniqueness. While it includes numerical experiments, the core contribution is in the advanced mathematical framework for minimum-deformation and moment matching. The empirical section demonstrates application rather than rigorous backtesting across diverse datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Construct Improved Comonotonic & Moment-Matching Approximations for Sums of Lognormals] --> B{Key Methodology: Minimum-Deformation Approach based on Step-Weighted Distribution};
    B --> C[Method Steps: 1. Start with comonotonic approximation. 2. Minimize Pearson deformation. 3. Apply moment-matching constraints. 4. Determine step quantiles/weights.];
    C --> D[Inputs: Sums of Lognormal Random Variables, Target Moment Orders (e.g., up to 5)];
    D --> E[Computational Processes: Joint Optimization Problem (reduced to partition points when steps = order + 1)];
    E --> F[Key Findings: 1. Existence & uniqueness of min-deformation comonotonic approximations established. 2. Framework accommodates moment matching of any finite order. 3. Numerical experiments up to 5 moments demonstrated. 4. Minimizing Pearson deformation doesn't guarantee improved risk measure accuracy (depends on starting approximation, partition points, correction weights).];