Fast simulation of Volterra processes using random Fourier features with application to the log-stationary fractional Brownian motion

TL;DR

This paper introduces a fast, accurate simulation method for Volterra processes using Random Fourier Features, with applications to fractional Brownian motion, enhancing efficiency and providing rigorous error bounds.

Contribution

It develops a novel accelerated simulation scheme based on spectral kernel representation and Hamiltonian Monte Carlo sampling, with theoretical guarantees and broader applicability than existing methods.

Findings

The method achieves high accuracy in kernel approximation.

It provides reliable parameter recovery in simulations.

The approach is computationally efficient and outperforms traditional methods.

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.