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On a Stationarity Theory for Stochastic Volterra Integral Equations

Emmanuel Gnabeyeu, Gilles Pagès

TL;DR

This work develops a stationarity theory for forward stochastic Volterra integral equations (SVIEs) with convolutive kernels by introducing a stabilizer that solves an intrinsic convolution equation. The key idea is to realize fake stationary regimes where either the first two moments are constant or the marginals are time-invariant in a Gaussian setting, without requiring true stationarity of the process. The authors derive moment equations, Wiener–Hopf representations, and L^p-confluence results that establish long-run convergence of shifted processes to a limiting (weakly) stationary regime, and they apply the framework to α-fractional and exponential-fractional kernels, deriving existence of stabilizers, providing asymptotics, and furnishing numerical illustrations. The results yield stabilized volatility models that capture short- and long-memory effects without relying on classical stationary dynamics, with implications for rough volatility modeling in finance. Overall, the paper provides a rigorous convolution-based approach to fake stationarity and long-run behavior for non-Markovian SVIEs, along with practical computational tools for fractional kernels.

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.

On a Stationarity Theory for Stochastic Volterra Integral Equations

TL;DR

This work develops a stationarity theory for forward stochastic Volterra integral equations (SVIEs) with convolutive kernels by introducing a stabilizer that solves an intrinsic convolution equation. The key idea is to realize fake stationary regimes where either the first two moments are constant or the marginals are time-invariant in a Gaussian setting, without requiring true stationarity of the process. The authors derive moment equations, Wiener–Hopf representations, and L^p-confluence results that establish long-run convergence of shifted processes to a limiting (weakly) stationary regime, and they apply the framework to α-fractional and exponential-fractional kernels, deriving existence of stabilizers, providing asymptotics, and furnishing numerical illustrations. The results yield stabilized volatility models that capture short- and long-memory effects without relying on classical stationary dynamics, with implications for rough volatility modeling in finance. Overall, the paper provides a rigorous convolution-based approach to fake stationarity and long-run behavior for non-Markovian SVIEs, along with practical computational tools for fractional kernels.

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 in the sense that all marginal distributions share the same expectation and variance. This effect is achieved by introducing a deterministic stabilizer associated with the kernel.We also look at the -confluence (for ) of such process as time goes to infinity(i.e. we investigate if its marginals when starting from various initial values are confluent in 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 -gamma fractional integration kernel, where enters the regime of whereas regularizes diffusion paths and invoke , persistence or long range dependence. With this fake stationary Volterra processes, we introduce a family of stabilized volatility models.

Paper Structure

This paper contains 31 sections, 24 theorems, 170 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Plan of the paper and Notations
  4. Background on Stochastic Volterra equations with convolutive kernels
  5. Volterra processes with convolutive kernels
  6. Fourier-Laplace transforms and Solvent core of a convolutive kernel
  7. Application to the Wiener-Hopf equation
  8. Investigating stationarity of a scaled stochastic Volterra Integral equation
  9. Towards stationarity of First Moments.
  10. Stationarity of the Mean
  11. Towards stationarity of the variance
  12. Stabilizer and Fake Stationary Regimes
  13. Examples of fake stationary regimes of type I and II
  14. Towards Long run behaviour: asymptotics and confluence
  15. Moments control.
  16. ...and 16 more sections

Key Result

Proposition 2.4

Let $g, h: \mathbb{R}_+ \to \mathbb{R}$ be two locally bounded Borel function, let $K \! \in L^1_{loc}(Leb_{\mathbb{R}_+})$ and let $\lambda \!\in \mathbb{R}$. Assume that the $\lambda$-resolvent $R_{\lambda}$ of $K$ is differentiable on $(0, +\infty)$ with a derivative $R'_{\lambda}\!\in L^1_{l

Figures (13)

  • Figure 1: Jordan contour $\Gamma_{\gamma, \delta, R}$.
  • Figure 2: Curves of $R_{\alpha,\lambda}(t)$ and $f_{\alpha,\lambda}(t)$ for different values of $\alpha \in [\frac{1}{2},1)$
  • Figure 3: Curves of $R_{\alpha,\lambda}(t)$ and $f_{\alpha,\lambda}(t)$ for different values of $\alpha \in (1,2)$
  • Figure 4: Graph of the stabilizer $t \to \varsigma_{\alpha,\lambda,c}(t)$ over time interval [0, T ], T = 10 for a value of the Hurst esponent $H=0.8$, $\lambda = 0.2$, c = 0.3.
  • Figure 5: Confluence from a [0,30]-Uniform Distribution, T=60, $H=0.8$, $\lambda = 0.2$, c = 0.36.
  • ...and 8 more figures

Theorems & Definitions (31)

  • Definition 2.1: Convolutive kernel and Volterra equations
  • Example 2.3: Laplace transform and $\lambda-$ Resolvent associated to the Exponential-fractional Kernel
  • Proposition 2.4: Wiener-Hopf and Resolvent equations
  • Lemma 3.1
  • Proposition 3.2: Wiener-Hopf transform
  • Remark 3.3
  • Proposition 3.4: Stationarity of the first moment
  • Theorem 3.5: Time-dependent or inhomogenous diffusion coefficient $\sigma$
  • Definition 3.6: Stationary of Order $p\geq1$ and Fake stationary regime of type I and II (see. Pages2024)
  • Definition 3.7
  • ...and 21 more