Single- and Multi-Level Fourier-RQMC Methods for Multivariate Shortfall Risk

2026-02-06

AI Summary

This study addresses the high computational cost and slow convergence inherent in optimal capital allocation under multivariate shortfall risk. It proposes a novel integration of Fourier inversion with randomized quasi-Monte Carlo (RQMC) methods to efficiently evaluate expectations arising in risk constraints and optimization within the frequency domain. By leveraging the smoothness of the integrand, the approach enhances integration efficiency and establishes both single-level and multi-level RQMC algorithmic frameworks. Empirical results demonstrate that the proposed method significantly outperforms sample average approximation and stochastic optimization benchmarks across diverse risk factor and loss structures, achieving higher accuracy, lower computational cost, and faster asymptotic convergence—thereby confirming its superior trade-off between complexity and efficiency.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• multivariate shortfall risk
• capital allocation
• numerical estimation
• computational cost
• Monte Carlo methods

Innovation

• Fourier inversion
• Randomized Quasi-Monte Carlo (RQMC)
• Multilevel methods
• Multivariate shortfall risk
• Frequency-domain integration