Paper: arXiv 2609.24548

Authors: Boris Günther, Ludger Overbeck

Abstract

We study affine stochastic Volterra equations on the cone of symmetric positive semidefinite matrices. For scalar kernels acting entrywise on the matrix dynamics, we establish weak existence by exploiting stochastic invariance results for Volterra equations on convex domains and derive a conditional Fourier–Laplace transform formula characterized by matrix-valued Riccati–Volterra equations. As an application, we extend the Gibson–Schwartz commodity model by replacing its variance-covariance structure with a Volterra–Wishart process. The resulting model allows for memory in the variances and for stochastic instantaneous correlation, while retaining affine tractability. Its joint Fourier–Laplace transform admits an exponential-affine representation governed by a matrix Riccati–Volterra equation. While the existence theory considered here excludes kernels that are singular at the origin, shifted fractional kernels remain admissible and provide a tractable specification with power-law memory.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly mathematical and novel extension of affine processes to Volterra equations on matrix cones, with a specific application to commodity markets. While the theoretical development is strong, the empirical rigor is limited to model formulation rather than extensive backtesting or data validation. The overall score reflects the significant theoretical contribution and novelty.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Gibson-Schwartz Model with Memory & Stochastic Correlation] --> B{Methodology: Affine Stochastic Volterra Equations on SPD Matrices};
    B --> C[Key Concepts: Stochastic Invariance, Conditional Fourier-Laplace Transform, Matrix Riccati-Volterra Equations];
    C --> D{Model Application: Volterra-Wishart Process for Commodity Markets};
    D --> E[Computational Processes: Derivation of Exponential-Affine Fourier-Laplace Transform];
    E --> F[Key Outcome 1: Weak Existence of Solutions for Affine Volterra Equations];
    E --> G[Key Outcome 2: Affine Tractability & Memory in Variances/Stochastic Instantaneous Correlation];
    E --> H[Key Outcome 3: Admissibility of Shifted Fractional Kernels for Power-Law Memory];