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

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.