On (fake) Stationarity in Stochastic Volterra Equations with Affine Drift and Regular Kernels

Abstract

We investigate the fake stationarity properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs) with affine drift and long-memory (regular) kernels, both on finite horizons and in the long-run regime. By either deriving explicit closed-form specifications for 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 $\mu$ fully flexible, we show that it is possible to induce a \textit{fake stationary} regime, in the sense that all marginal distributions share the same mean and variance. Afterwards, using a refined asymptotic analysis, we further establish that, in both frameworks, the time-shifted solutions of these long-memory SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance structure, for suitable classes of diffusion coefficients. These results are applied to a class of exponential-fractional Stochastic Volterra Integral Equations driven by an $\alpha$-gamma fractional integration kernel, in the particular regime \(\alpha \geq 1\), which regularizes diffusion paths and invoke textit{ long-term memory}, persistence or long range dependence.