Asymptotic behavior of the multilevel type error for SDEs driven by a pure jump L\'evy process

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.