  In this paper, we analyze the drift-implicit (or backward) Euler numerical
scheme for a class of stochastic differential equations with unbounded drift
driven by an arbitrary $\lambda$-H\"older continuous process,
$\lambda\in(0,1)$. We prove that, under some mild moment assumptions on the
H\"older constant of the noise, the $L^r(\Omega;L^\infty([0,T]))$-rate of
convergence is equal to $\lambda$. To exemplify, we consider numerical schemes
for the generalized Cox--Ingersoll-Ross and Tsallis--Stariolo--Borland models.
The results are illustrated by simulations.
