AI 中文总结
研究由埃尔米特过程驱动的广义CIR扩散，利用其赫尔德正则性在杨氏 - 斯蒂尔杰斯意义下解释动力学，建立了适定性、正性界、Malliavin可微性及绝对连续性四个主要结果。
AI 中文摘要
我们研究广义Cox--Ingersoll--Ross（CIR）扩散 \begin{equation*} dX_t = a\bigl(b(t)-X_t\bigr)\,dt + \bigl(\sigma_0+\sigma_1\sqrt{\phi_\eps(X_t)}\bigr)\,dZ_t^{(q,H)}, \quad X_0 = x_0 > 0, \end{equation*} 其中 $Z^{(q,H)}$ 是阶数 $q\geq 1$ 且赫斯特参数 $H\in(1/2,1)$ 的埃尔米特过程，$\phi_\eps$ 是平方根的光滑正则化。该框架同时捕捉长程相依、非高斯创新（$q\geq 2$ 时）以及经典CIR模型的正性。由于 $q\geq 2$ 的埃尔米特过程不是半鞅，我们在杨氏 - 斯蒂尔杰斯意义下逐路径解释动力学，利用 $Z^{(q,H)}$ 的赫尔德正则性。我们建立了四个主要结果。首先是适定性：在全局利普希茨系数（$\phi_\eps$ 满足）下，分数阶索伯列夫空间中存在唯一强解。其次是定量正性界：当 $\sigma_1 = 0$ 时，$X$ 在 $[0,T]$ 上保持为正的概率由一个显式表达式下界界定，当初始水平和长期目标相对于 $\sigma_0$ 变大时趋于 1；这里未建立经典布朗情形下的几乎必然陈述，因为通常的边界不可达机制依赖于非半鞅驱动不可用的工具。第三是 Malliavin 可微性：$X_t\in\D^{1,\infty}$，Malliavin 导数有显式公式，是线性化杨氏随机微分方程的解。第四是绝对连续性：对所有 $t>0$，$X_t$ 的分布关于勒贝格测度绝对连续。
英文摘要
We study a generalized Cox--Ingersoll--Ross (CIR) diffusion \begin{equation*} dX_t = a\bigl(b(t)-X_t\bigr)\,dt + \bigl(σ_0+σ_1\sqrt{ϕ_\eps(X_t)}\bigr)\,dZ_t^{(q,H)}, \quad X_0 = x_0 > 0, \end{equation*} where $Z^{(q,H)}$ is a Hermite process of order $q\geq 1$ and Hurst parameter $H\in(1/2,1)$, and $ϕ_\eps$ is a smooth regularisation of the square root. This framework simultaneously captures long-range dependence, non-Gaussian innovations (for $q\geq 2$), and the positivity of the classical CIR model. Since Hermite processes with $q\geq 2$ are not semimartingales, we interpret the dynamics pathwise in the Young--Stieltjes sense, exploiting the Hölder regularity of $Z^{(q,H)}$. We establish four main results. First, well-posedness: a unique strong solution exists in a fractional Sobolev space, under globally Lipschitz coefficients (satisfied by $ϕ_\eps$). Second, a quantitative positivity bound: for $σ_1=0$, the probability that $X$ stays positive on $[0,T]$ is bounded below by an explicit expression tending to $1$ as the initial level and long-run target grow large relative to $σ_0$; an almost-sure statement, available in the classical Brownian case, is not established here, since the usual boundary-non-attainment mechanism relies on tools unavailable for a non-semimartingale driver. Third, Malliavin differentiability: $X_t\in\D^{1,\infty}$, with an explicit formula for the Malliavin derivative as the solution of a linearised Young SDE. Fourth, absolute continuity: the law of $X_t$ is absolutely continuous with respect to the Lebesgue measure for all $t>0$.
