Absolute continuity of finite-dimensional distributions of Hermite processes via Malliavin calculus

We investigate the existence of densities for finite-dimensional distributions of Hermite processes of order $q\ge 1$ and self-similarity parameter $H\in (\frac{1}{2},1)$. Whereas the Gaussian case $q=1$ (fractional Brownian motion) is well understood, the non-Gaussian situation has not yet been settled. In this work, we extend the classical three-step approach used in the Gaussian case: factorization of the determinant into conditional terms, strong local nondeterminism, and non-degeneracy. We transport this strategy to the Hermite setting using Malliavin calculus. Specifically, we establish a determinant identity for the Malliavin matrix, prove strong local nondeterminism at the level of Malliavin derivatives, and apply the Bouleau-Hirsch criterion. Consequently, for any distinct times $t_1,\dots ,t_n$, the vector $(Z_{t_1}^{H,q},\dots ,Z_{t_n}^{H,q})$ of a Hermite process admits a density with respect to the Lebesgue measure. Beyond the result itself, the main contribution is the methodology, which could extend to other non-Gaussian models.