Affine Volterra covariance processes and application to commodity markets

Boris G\"unther and Ludger Overbeck

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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.

Date: 2026-09
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