Paper: arXiv 2603.02946

Authors: Othmane Zarhali, Nicolas Langrené

Abstract

A fast simulation framework for stochastic Volterra processes based on Random Fourier Features (RFF) approximation of the kernel is developed. After recalling the main properties of Volterra processes and reviewing existing numerical simulation methods, an accelerated scheme is introduced that relies on a spectral representation of the kernel. A particular attention is devoted to sampling from the kernel spectral density using Hamiltonian Monte Carlo, whose efficiency and stability bring more convenience than alternative sampling procedures. Quantitative guarantees for the proposed method are established, including moment estimates and strong error bounds. The approach is further compared with the kernel approximation by sum of exponentials (Random Laplace Features) commonly used in the literature, emphasizing the broader generality of the present framework. As a primary application, Volterra processes associated with the Stationary fractional Brownian Motion (S-fBM) kernel are investigated. A spectral density representation is derived in closed form using hypergeometric functions, a condition for positive definiteness is established, and explicit truncation as well as Monte Carlo error bounds are provided for the RFF approximation in this setting. Numerical experiments in dimensions one and two illustrate the accuracy of the kernel approximation, the reliable recovery of model parameters, and the competitiveness of the accelerated simulation scheme in terms of computational efficiency and both weak and strong error performance.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework for simulating Volterra processes via Random Fourier Features, establishing error bounds and hypergeometric closed-form solutions. The empirical section covers numerical simulations in 1D/2D demonstrating weak/strong error performance and parameter recovery, though it focuses primarily on synthetic simulation experiments rather than extensive empirical asset pricing backtests.

Research Flowchart

  flowchart TD
    A[Goal: Fast & accurate simulation of Stochastic Volterra Processes] --> B[Inputs: Kernel K, Hurst index H, Dimension d]
    
    B --> C[Methodology: Random Fourier Features RFF Kernel Approximation]
    
    C --> D[Compute Spectral Density via Hypergeometric Functions]
    
    D --> E[Sample Spectral Frequencies via Hamiltonian Monte Carlo HMC]
    
    E --> F[Simulate Volterra Process & Log-Stationary fBM]
    
    F --> G[Outcomes: Strong/Weak Error Guarantees, Parameter Recovery & Computational Efficiency]