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 ≥ 1 q \ge 1 and self-similarity parameter H ∈ ( 1 2 , 1 ) H\in (\frac 12,1) . Whereas the Gaussian case q = 1 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 , … , t n t_1,\dots ,t_n , the vector ( Z t 1 H , q , … , Z t n H , q ) (Z^{H,q}_{t_1},\dots ,Z^{H,q}_{t_n}) 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.