Paper: arXiv 2504.19885

Authors: Eduardo Abi Jaber, Elie Attal

Abstract

We introduce a novel simulation scheme, iVi (integrated Volterra implicit), for integrated Volterra square-root processes and Volterra Heston models based on the Inverse Gaussian distribution. The scheme is designed to handle $L^1$ kernels with singularities by relying solely on integrated kernel quantities, and it preserves the non-decreasing property of the integrated process. We establish weak convergence of the iVi scheme by reformulating it as a stochastic Volterra equation with a measure kernel and proving a stability result for this class of equations. Numerical results demonstrate that convergence is achieved with very few time steps. Remarkably, for the rough fractional kernel, unlike existing schemes, convergence seems to improve as the Hurst index $H$ decreases and approaches $-1/2$.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a breakthrough numerical simulation scheme (iVi) for integrated Volterra square-root processes using Inverse Gaussian sampling, featuring rigorous weak convergence proofs on L1 kernels. It combines deep theoretical stochastic analysis with impressive practical performance, demonstrating unprecedented fast convergence in hyper-rough volatility regimes (H down to -0.49).

Research Flowchart

  flowchart TD
    A[Research Goal: Simulate Volterra square-root processes & Volterra Heston models] --> B[Inputs: L1 kernels with singularities & Inverse Gaussian distribution]
    B --> C[Methodology: Develop iVi Scheme using integrated kernel quantities]
    C --> D[Theory: Prove Weak Convergence via Stochastic Volterra Equations]
    D --> E[Computation: Run simulations across time steps and Hurst index H]
    E --> F[Findings: Robust convergence with few steps; accuracy improves as H → -1/2]

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4 of 54 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.