带仿射漂移与正则核的随机沃尔泰拉方程的（伪）平稳性研究
On (fake) Stationarity in Stochastic Volterra Equations with Affine Drift and Regular Kernels
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中文总结 AI 辅助
研究带仿射漂移与正则核的随机沃尔泰拉方程的伪平稳性，通过两种方法诱导伪平稳 regime，证明时移解弱收敛至L²平稳过程，并将结果应用于α≥1的指数-分数SVIE。
中文摘要 AI 辅助
我们研究带仿射漂移与长记忆（正则）核的前向随机沃尔泰拉积分方程（SVIEs）解的伪平稳性性质，分别针对有限时间范围与长期 regime。通过推导漂移项中确定性初值φ和均值回复函数μ的显式闭式表达式，或在与核关联的扩散系数中引入确定性稳定因子ς同时保持μ完全灵活，可诱导出“伪平稳” regime，其所有边际分布具有相同均值与方差。随后，通过精细渐近分析进一步证明，在两类框架下，对于合适的扩散系数类，这些长记忆SVIE的时移解在泛函意义上弱收敛至一族L²平稳过程，且该族过程具有相同协方差结构。上述结果应用于一类由α-伽马分数积分核驱动的指数-分数随机沃尔泰拉积分方程，在α≥1的特定 regime 下，该核可正则化扩散路径并引入长记忆、持续性或长程相关性。
英文摘要
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.
发表机构
- Laboratoire de Probabilités, Statistique et Modélisation (LPSM), UMR 8001（概率、统计与建模实验室（LPSM））
- Sorbonne Université（索邦大学）
- Université Paris Cité（巴黎西岱大学）
机构由 AI 辅助整理，请以论文原文为准。
