  Motivated by the multilevel Monte Carlo method introduced by Giles [5], we
study the asymptotic behavior of the normalized error process
$u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are respectively Euler
approximations with time steps $1/n$ and $1/nm$ of a given stochastic
differential equation $X$ driven by a pure jump L\'evy process. In this paper,
we prove that this normalized multilevel error converges to different
non-trivial limiting processes with various sharp rates $u_{n,m}$ depending on
the behavior of the L\'evy measure around zero. Our results are consistent with
those of Jacod [9] obtained for the normalized error $u_n(X^n-X)$, as when
letting $m$ tends to infinity, we recover the same limiting processes. For the
multilevel error, the proofs of the current paper are challenging since unlike
[9] we need to deal with $m$ dependent triangular arrays instead of one.
