On a Stationarity Theory for Stochastic Volterra Integral Equations

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 $\varsigma$ 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 $\alpha$-gamma fractional integration kernel, where $\alpha\leq 1$ enters the regime of $\textit{rough path}$ whereas $\alpha> 1$ 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.