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];

Cited References (41)

  1. Affine Volterra Processes (2019)
  2. Weak Solutions of Stochastic Volterra Equations in Convex Domains with General Kernels (2026)
  3. Inhomogeneous Affine Volterra Processes (2022)
  4. Dynamic Roughness in the Term Structure of Oil Markets Volatility (2023)
  5. Forecasting Commodity Markets Volatility : HAR or Rough ? (2020)
  6. Forecasting Volatility in Commodity Markets with Long-Memory Models (2022)
  7. Nonnegativity Preserving Convolution Kernels. Application to Stochastic Volterra Equations in Closed Convex Domains and Their Approximation — → 2302.07758
  8. Volatility Spillovers in Commodity Markets: A Large t-Vector Autoregressive Approach (2020)
and 33 more
  1. A Rough Multi-Factor Model of Electricity Spot Prices (2017)
  2. Convenience Yields for CO2 Emission Allowance Futures Contracts (2006)
  3. Wishart Processes (1991)
  4. Pricing Commodity Futures Options in the Schwartz Multi Factor Model with Stochastic Volatility: An Asymptotic Method (2017)
  5. A Theory of the Term Structure of Interest Rates (1985) — link
  6. Affine Processes on Positive Semidefinite Matrices (2011)
  7. Affine Processes on Symmetric Cones (2016)
  8. Markovian Lifts of Positive Semidefinite Affine Volterra-type Processes (2019)
  9. Rough Volatility Dynamics in Commodity Markets (2026) — link
  10. Stochastic Models of Energy Commodity Prices and Their Applications : Mean-reversion with Jumps and Spikes (2000)
  11. Better to Give than to Receive: Predictive Directional Measurement of Volatility Spillovers (2012)
  12. Measuring Financial Asset Return and Volatility Spillovers , with Application to Global Equity Markets (2009)
  13. Affine Processes and Applications in Finance (2003)
  14. Volatility Is Rough (2018)
  15. Stochastic Convenience Yield and the Pricing of Oil Contingent Claims (1990)
  16. Continuous Time Wishart Process for Stochastic Risk (2006)
  17. Wishart Quadratic Term Structure Models (2003)
  18. Volterra Integral and Functional Equations (1990)
  19. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993)
  20. Asymptotic Pricing of Commodity Derivatives Using Stochastic Volatility Spot Models (2008)
  21. A Maximal Affine Stochastic Volatility Model of Oil Prices (2010)
  22. The Laplace Transform of the Integrated Volterra Wishart Process (2022) — link
  23. Does Economic Policy Uncertainty Drive Volatility Spillovers in Electricity Markets: Time and Frequency Evidence (2022)
  24. Fractional Brownian Motions , Fractional Noises and Applications (1968)
  25. Convenience Yield and the Theory of Storage: Applying an Option-Based Approach (2015)
  26. Volatility and Commodity Price Dynamics (2004)
  27. Stochastic Volatility and Seasonality in Commodity Futures and Options: The Case of Soybeans (2002)
  28. Equilibrium Forward Curves for Commodities (2000)
  29. The Impact of Renewables on Spillover Effects in Electricity Markets (2025)
  30. Revisiting the Gibson-Schwartz and Schwartz-Smith Commodity Models (2025)
  31. Short- Term Variations and Long-Term Dynamics in Commodity Prices (2000)
  32. Convenience Yields and Risk Premiums in the EU - ETS — Evidence from the Kyoto Commitment Period (2016)
  33. The Stochastic Fubini Theorem Revisited (2012)

1 of 41 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.