On a Stationarity Theory for Stochastic Volterra Integral Equations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918187321262080 |
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| author | Gnabeyeu, Emmanuel Pagès, Gilles |
| author_facet | Gnabeyeu, Emmanuel Pagès, Gilles |
| contents | 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 $ς$ 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 $α$-gamma fractional integration kernel, where $α\leq 1$ enters the regime of $\textit{rough path}$ whereas $α> 1$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03474 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Stationarity Theory for Stochastic Volterra Integral Equations Gnabeyeu, Emmanuel Pagès, Gilles Probability Dynamical Systems Mathematical Finance 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 $ς$ 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 $α$-gamma fractional integration kernel, where $α\leq 1$ enters the regime of $\textit{rough path}$ whereas $α> 1$ 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. |
| title | On a Stationarity Theory for Stochastic Volterra Integral Equations |
| topic | Probability Dynamical Systems Mathematical Finance |
| url | https://arxiv.org/abs/2511.03474 |