On a Stationarity Theory for Stochastic Volterra Integral Equations with Affine Drift

Emmanuel Gnabeyeu and Gilles Pag\`es

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Abstract: This paper investigate the properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs for short) with affine drift, specifically their stationarity, both over a finite horizon and in the long run. We demonstrate 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 phenomenon can be achieved either through explicit closed-form specifications of the deterministic initial condition $\phi$ and the mean-reversion function $\mu$ appearing in the drift, or by introducing a deterministic stabilizing factor $\varsigma$ in the diffusion coefficient, associated with the kernel, while keeping the function $\mu$ otherwise fully flexible. We further look at the $L^p$-confluence properties $p>0$ of such processes as time goes to infinity, namely we investigate whether the marginals of solutions associated with different initial values become asymptotically confluent in $L^p$. We finally study the functional weak long-run asymptotics for some classes of diffusion coefficients. More precisely, we establish that, in both settings, the time-shifted solutions of such SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance function. These results are then applied to a class of Exponential-Fractional Stochastic Volterra Integral Equations driven by an $\alpha$-gamma fractional integration kernel,in the particular case $\alpha \in (0,1]$, which corresponds to the $\textit{rough-path}$ regime. Building on these fake stationary Volterra processes, we finally introduce a family of stabilized Rough volatility models.

Date: 2025-11, Revised 2026-09
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