[论文解读] Single- and Multi-Level Fourier-RQMC Methods for Multivariate Shortfall Risk

Chiheb Ben Hammouda, Truong Nguyen|arXiv (Cornell University)|Feb 6, 2026

Risk and Portfolio Optimization被引用 0

一句话总结

该论文开发了一水平和多水平的 Fourier-RQMC 方法，用于高效估计多元短缺风险（multivariate shortfall risk）及相应资本分配，利用傅里叶反演和 RQMC 实现更好的收敛性和计算复杂度。

ABSTRACT

Multivariate shortfall risk measures provide a principled framework for quantifying systemic risk and determining capital allocations prior to aggregation in interconnected financial systems. Despite their well established theoretical properties, the numerical estimation of multivariate shortfall risk and the corresponding optimal allocations remains computationally challenging, as existing Monte Carlo based approaches can be numerically expensive due to slow convergence. In this work, we develop a new class of single and multilevel numerical algorithms for estimating multivariate shortfall risk and the associated optimal allocations, based on a combination of Fourier inversion techniques and randomized quasi Monte Carlo (RQMC) sampling. Rather than operating in physical space, our approach evaluates the relevant expectations appearing in the risk constraint and its optimization in the frequency domain, where the integrands exhibit enhanced smoothness properties that are well suited for RQMC integration. We establish a rigorous mathematical framework for the resulting Fourier RQMC estimators, including convergence analysis and computational complexity bounds. Beyond the single level method, we introduce a multilevel RQMC scheme that exploits the geometric convergence of the underlying deterministic optimization algorithm to reduce computational cost while preserving accuracy. Numerical experiments demonstrate that the proposed Fourier RQMC methods outperform sample average approximation and stochastic optimization benchmarks in terms of accuracy and computational cost across a range of models for the risk factors and loss structures. Consistent with the theoretical analysis, these results demonstrate improved asymptotic convergence and complexity rates relative to the benchmark methods, with additional savings achieved through the proposed multilevel RQMC construction.

研究动机与目标

• 推进多变量短缺风险度量（MSRM）及相应分配的数值估计。
• 结合傅里叶反演与随机化拟蒙特卡洛（RQMC），利用频域积分项的平滑性。
• 对单级与多级方案提供严格的误差与计算复杂度分析。
• 通过自适应阻尼和域变换保持优化轨迹上被积函数的正则性。
• 开发一个可扩展的优化框架（基于 SQP），整合傅里叶-RQMC 代理。

提出的方法

• 在频域通过损失函数及其导数的傅里叶变换来表示 MSRM 的期望与梯度。
• 使用可容忍的等高线移位（阻尼）以确保傅里叶积分项的可积性和光滑性。
• 将高维傅里叶积分分解为基于相互作用阶数的有限个低维分量积分之和。
• 在 SQP 框架内应用单级傅里叶–RQMC 来估计 g(m)、∇g(m)、∇²g(m)。
• 扩展到利用优化几何收敛来降低成本的多级傅里叶–RQMC 方案。
• 用带有阻尼感知的线搜索和 SLSQP 实现来求解得到的 SQP 子问题。

实验结果

研究问题

• RQ1 是否可以使用频域表示和 RQMC 抽样有效估计 MSRM 及相关最优分配？
• RQ2 阻尼规则和域变换如何影响基于傅里叶的 MSRM 估计量的正则性和准确性？
• RQ3 多级 RQMC 构造在保持优化轨迹准确性的同时是否降低计算成本？
• RQ4 单级与多级傅里叶–RQMC 在估计 MSRM 时的收敛性与复杂度性质如何？
• RQ5 在不同损失结构和维度下，这些方法与 SAA 与 SA 基准相比有何表现？

主要发现

• 与基准 Monte Carlo 方法相比，傅里叶–RQMC 估计量提供更好的渐近收敛性和复杂度。
• 多级 RQMC 构造通过利用优化的几何收敛性在降低计算成本方面实现了额外的改进。
• 自适应阻尼和正则化更新规则在优化轨迹上保持积分的鲁棒性。
• 针对 RQMC 的域变换有助于保持正则性并改进积分过程中的边界处理。
• 数值实验表明在各种损失模型和维度下，傅里叶–RQMC 相较标准基准具有更优表现。