On a Stationarity Theory for Stochastic Volterra Integral Equations

TL;DR

This paper analyzes the stationarity and long-term behavior of stochastic Volterra integral equations, revealing conditions for fake stationarity and exploring implications for stabilized volatility models with memory effects.

Contribution

It introduces the concept of fake stationarity in non-Markovian systems and demonstrates how to induce it using a deterministic stabilizer, advancing understanding of long-term dynamics.

Findings

Fake stationary regime can be induced with a stabilizer.

Solutions do not exhibit strong stationarity unless kernel is constant.

Long-term behavior varies with fractional kernel parameters.

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.