We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. As a main difference compared to Gobet-Landon, Annals of Applied Probability 2014 (DOI: 10.1214/13-AAP959), the martingale part of the semi-martingale may be degenerate. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand.