  We show that the rate of convergence of asymptotic expansions for solutions
of SDEs is generally higher in the case of degenerate (or partial) diffusion
compared to the elliptic case, i.e. it is higher when the Brownian motion
directly acts only on some components of the diffusion. In the scalar case,
this phenomenon was already observed in (Gobet and Miri 2014) using Malliavin
calculus techniques. In this paper, we provide a general and detailed analysis
by employing the recent study of intrinsic functional spaces related to
hypoelliptic Kolmogorov operators in (Pagliarani et al. 2016). Relevant
applications to finance are discussed, in particular in the study of
path-dependent derivatives (e.g. Asian options) and in models incorporating
dependence on past information.
