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On (fake) Stationarity in Stochastic Volterra Equations with Affine Drift and Regular Kernels
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- OpenAlex
- DOI
- 10.48550/arxiv.2608.31099
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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 $ϕ$ and the mean-reversion function $μ$ appearing in the drift, or by introducing a deterministic stabilizing factor $ς$ in the diffusion coefficient associated with the kernel while keeping $μ$ 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 $α$-gamma fractional integration kernel, in the particular regime \(α\geq 1\), which regularizes diffusion paths and invoke textit{ long-term memory}, persistence or long range dependence.
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