Paper: arXiv 2512.09590

Authors: Emmanuel Gnabeyeu, Gilles Pagès, Mathieu Rosenbaum

Abstract

True Volterra equations are inherently non stationary and therefore do not admit $\textit{genuine stationary regimes}$ over finite horizons. This motivates the study of the finite-time behavior of the solutions to scaled inhomogeneous affine Stochastic Volterra equations through the lens of a weaker notion of stationarity referred to as $\textit{fake stationary regime}$ in the sense that all marginal distributions share the same expectation and variance. As a first application, we introduce the $\textit{Fake stationary Volterra Heston model}$ and derive a closed-form expression for its characteristic function. Having established this finite-time proxy for stationarity, we then investigate the asymptotic (long-time) behavior to assess whether genuine stationary regimes emerge in the limit. Using an extension of the exponential-affine transformation formula for those processes, we establish in the long run the existence of limiting distributions, which (unlike in the case of classical affine diffusion processes) may depend on the initial state of the process, unless the Volterra kernel coincides with the $α-$ fractional integration kernel, for which the dependence on the initial state vanishes. We then proceed to the construction of stationary processes associated with these limiting distributions. However, the dynamics in this long-term regime are analytically intractable, and the process itself is not guaranteed to be stationary in the classical sense over finite horizons. This highlights the relevance of finite-time analysis through the lens of the aforementioned $\textit{fake stationarity}$, which offers a tractable approximation to stationary behavior in genuinely non-stationary Volterra systems.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous mathematical treatment of inhomogeneous affine Stochastic Volterra equations, introducing a new concept of ‘fake stationarity’ and deriving closed-form characteristic functions for the fake stationary rough Heston model. While the theoretical derivations, functional equations, and limit theorems are mathematically sophisticated and complete, empirical validation and real-world backtesting are minimal in the excerpt.

Research Flowchart

  flowchart TD
    A["Research Goal: Analyze Stationarity & Dynamics of Inhomogeneous Affine Volterra Processes"] --> B["Data/Inputs: Scaled Inhomogeneous Affine SDEs & Fractional Kernels"]
    B --> C["Finite-Time Analysis: Define 'Fake Stationarity' (Matching Expectation & Variance)"]
    C --> D["Application: Derive Closed-Form Characteristic Function for Fake Stationary Volterra Heston"]
    B --> E["Asymptotic Analysis: Apply Extended Exponential-Affine Formula for Long-Time Limits"]
    E --> F["Key Outcome: Limiting Distributions Depend on Initial State (Except for α-Fractional Kernel)"]
    D --> G["Final Conclusion: Fake Stationarity Provides Tractable Finite-Time Proxy for Non-Stationary Volterra Systems"]
    F --> G