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arXiv 2511.03474math.PRmath.DSq-fin.MF
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带仿射漂移的随机Volterra积分方程的平稳性理论

On a Stationarity Theory for Stochastic Volterra Integral Equations with Affine Drift

Emmanuel Gnabeyeu, Gilles Pagès

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43 pages,8 figures
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AI总结：

本文研究带仿射漂移的SVIE的平稳性，提出伪平稳状态的实现方式，证明其解的渐近汇聚与弱收敛性质，并将结果应用于指数-分数SVIE，引入稳定化粗糙波动率模型。

摘要：

本文研究带仿射漂移的前向随机Volterra积分方程（简称SVIE）解的性质，包括其有限时间范围和长期的平稳性。我们证明可以诱导出一种“伪平稳状态”，即所有边际分布具有相同的期望和方差，该现象可通过两种方式实现：一是明确指定漂移项中出现的确定性初始条件φ和均值回复函数μ的闭式形式；二是在扩散系数中引入与核相关的确定性稳定因子ς，同时保持函数μ完全灵活。我们进一步研究这类过程当时间趋于无穷时的L^p汇聚性质（p>0），即探讨与不同初始值相关的解的边际分布是否在L^p中渐近汇聚。最后，我们研究某些类别的扩散系数的泛函弱长期渐近性，更准确地说，我们证明在上述两种设定下，这类SVIE的时移解在泛函意义上弱收敛到一族L^2平稳过程，这些过程共享相同的协方差函数。这些结果随后被应用于一类由α-伽马分数积分核驱动的指数-分数随机Volterra积分方程，其中α∈(0,1]，对应“粗糙路径” regime。基于这些伪平稳Volterra过程，我们最终引入了一族稳定化粗糙波动率模型。

英文摘要：

This paper investigate the properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs for short) with affine drift, specifically their stationarity, both over a finite horizon and in the long run. We demonstrate 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 phenomenon can be achieved either through explicit closed-form specifications of 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 the function $μ$ otherwise fully flexible. We further look at the $L^p$-confluence properties $p>0$ of such processes as time goes to infinity, namely we investigate whether the marginals of solutions associated with different initial values become asymptotically confluent in $L^p$. We finally study the functional weak long-run asymptotics for some classes of diffusion coefficients. More precisely, we establish that, in both settings, the time-shifted solutions of such SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance function. These results are then applied to a class of Exponential-Fractional Stochastic Volterra Integral Equations driven by an $α$-gamma fractional integration kernel,in the particular case $α\in (0,1]$, which corresponds to the $\textit{rough-path}$ regime. Building on these fake stationary Volterra processes, we finally introduce a family of stabilized Rough volatility models.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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