2026/07/26 by Atef Lechiheb
#math.PR
We study a generalized Cox--Ingersoll--Ross (CIR) diffusion dXt = a(b(t)-Xt) dt + (σ0+σ1√(ϕ_\eps(Xt))) dZt(q,H), X0 = x0 > 0, where Z(q,H) is a Hermite process of order q≥ 1 and Hurst parameter H∈(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≥ 2), and the positivity of the classical CIR model. Since Hermite processes with q≥ 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 σ1=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 σ0; 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: Xt∈\D1,∞, with an explicit formula for the Malliavin derivative as the solution of a linearised Young SDE. Fourth, absolute continuity: the law of Xt is absolutely continuous with respect to the Lebesgue measure for all t>0.