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

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.

Figures (9)

• Figure 1: S-fBM kernel approximation using the RFF representation for different correlation limits $T$ together with the absolute error between the approximated and theoretical kernel. The number of spectral Monte Carlo simulations is $M = 8000$, with $\nu^2 = 50$ and $H = 0.1$. The top panels correspond to $T = 10$ and $T = 100$, while the bottom panel corresponds to $T = 200$.
• Figure 2: Two dimensional visualization of the S-fBM kernel approximation error obtained using the RFF representation. Results are based on $M = 8000$ Monte Carlo simulations with parameters $\nu^2 = 50$ and $H = 0.1$. The top panel shows the error surface, while the bottom panel displays the corresponding colormap both for $T = 100$.
• Figure 3: Using the RFF representation of the S-fBM kernel for different sampling realizations $M$, we plot the mean value and the associated $95\%$ confidence interval from a sample of $20$ copies of the Hurst exponent (blue) and the estimated intermittency parameter (red) together with error bars compared with the respective theoretical values: $H = 0.1,\lambda^2=0.02$ (bottom) $H = 0.2,\lambda^2=0.05$ (middle) and $H = 0.3,\lambda^2=0.1$ (top). Here $T=200$ and the time lag sequence used is $\{\tau_k=\lfloor 2^{\frac{k}{2}} \rfloor,k=0...,Q\}$ where $Q=19$.
• Figure 4: Sample paths generated using the Euler scheme with VS without the RFF approximation, driven by the same underlying Brownian path to facilitate comparison, with $H=0.1$ (left) and $H=0.2$ (right). The parameters considered are $\lambda^2 = 0.01$, $T = 100$ and $M = 10{,}000$. The volatility function is given by $\sigma(t,x) = 0.3\,(1 + 0.1x)$. The simulation uses $1000$ time steps over the interval $[0,1]$.
• Figure 5: Average elapsed time over 20 runs as a function of the number of time steps $N$ for the classical Euler scheme (green) and the RFF-Euler scheme, respectively, both including (purple) and excluding (black) the random features sampling. The parameters considered are $H = 0.1$, $\lambda^2 = 0.01$, $T = 100$ and $M = 8000$. The volatility function is given by $\sigma(t,x) = 0.3\,(1 + 0.1x)$. This is performed within a CPU-only job on a single node with 10 cores.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Proposition 1
• Definition 1
• Proposition 2
• Theorem 3
• Proposition 3
• Theorem 4
• proof
• Theorem 5