Paper: arXiv 2511.03474

Authors: Emmanuel Gnabeyeu, Gilles Pagès

Abstract

This paper provide a comprehensive analysis of the finite and long time behavior of continuous-time non-Markovian dynamical systems, with a focus on the forward Stochastic Volterra Integral Equations(SVIEs).We investigate the properties of solutions to such equations specifically their stationarity, both over a finite horizon and in the long run. In particular, we demonstrate that such an equation does not exhibit a strong stationary regime unless the kernel is constant or in a degenerate settings. However, we show that it is possible to induce a $\textit{fake stationary regime}$ in the sense that all marginal distributions share the same expectation and variance. This effect is achieved by introducing a deterministic stabilizer $ς$ associated with the kernel.We also look at the $L^p$ -confluence (for $p>0$) of such process as time goes to infinity(i.e. we investigate if its marginals when starting from various initial values are confluent in $L^p$ as time goes to infinity) and finally the functional weak long-run assymptotics for some classes of diffusion coefficients. Those results are applied to the case of Exponential-Fractional Stochastic Volterra Integral Equations, with an $α$-gamma fractional integration kernel, where $α\leq 1$ enters the regime of $\textit{rough path}$ whereas $α> 1$ regularizes diffusion paths and invoke $\textit{long-term memory}$, persistence or long range dependence. With this fake stationary Volterra processes, we introduce a family of stabilized volatility models.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework analyzing the asymptotic behavior and stationarity properties of Stochastic Volterra Integral Equations (SVIEs). It introduces the concept of a ‘fake stationary regime’ with high theoretical precision, though it is entirely theoretical without empirical backtesting or real-market data validation.

Research Flowchart

  flowchart TD
    A["Research Question: How to establish stationarity & long-run asymptotics for SVIEs?"] --> B["Input: Forward SVIEs & α-Gamma Fractional Kernels"]
    B --> C["Theoretical Analysis: Strong Stationarity & L^p-Confluence"]
    C --> D["Methodology: Introduce Deterministic Stabilizer ς for 'Fake Stationarity'"]
    D --> E["Application: Exponential-Fractional SVIEs (Rough α ≤ 1 & Long-Memory α > 1)"]
    E --> F["Outcomes: Proof of Strong Stationarity Absence, Fake Stationary Regime & Stabilized Volatility Models"]