A CIR-Type Diffusion Driven by Hermite Processes: Well-Posedness, Positivity and Malliavin Analysis
We study a generalized Cox--Ingersoll--Ross (CIR) diffusion \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*} where $Z^{(q,H)}$ is a Hermite process of order $q\geq 1$ and Hurst parameter $H\in(1/2,1)$, and $\phi_\eps$ 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 H\"{o}lder 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 $\phi_\eps$). Second, a quantitative positivity bound: for $\sigma_1=0$, 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 $\sigma_0$; 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 $t>0$.