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

TL;DR

This paper analyzes the asymptotic behavior of multilevel error in Euler approximations of SDEs driven by pure jump Lévy processes, revealing different limiting processes depending on the Lévy measure.

Contribution

It extends previous work by studying the multilevel error's asymptotics, accounting for the dependence on the multilevel parameter m, and establishing convergence to various limiting processes.

Findings

Normalized multilevel error converges to different non-trivial limits.

Rates of convergence depend on the Lévy measure near zero.

Results align with previous single-level error analyses as m increases.

Abstract

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.