Asymptotic behavior of a multilevel type error for SDEs driven by a pure jump Lévy process
M. Ben Alayaa,
A. Kebaierb,
T. T.-B. Ngôc a Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, Saint-Etienne-du-Rouvray, France
b Laboratoire de Mathématiques et Modélisation d'Evry, CNRS, Université d'Evry, Université Paris-Saclay, Evry, France
c Laboratoire Manceau de Mathématiques, Le Mans Université, Le Mans, France
Abstract:
Motivated by the multilevel Monte Carlo method introduced by Giles [
Oper. Res., 56 (2008), pp. 607–617], 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 driven by a pure jump Lévy process. In this paper, we prove that this normalized multilevel error converges to different nontrivial limiting processes with various sharp rates
$u_{n,m}$ depending on the behavior of the Lévy measure around zero. Our results are consistent with those of Jacod [
Ann. Probab., 32 (2004), pp. 1830–1872] 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 Jacod's paper, we need to deal with
$m$ dependent triangular arrays instead of one.
Keywords:
Euler discretization, stochastic differential equations, pure jump Lévy processes, limit theorems, multilevel type error.
MSC: Primary 60J75,
65C30;
secondary 60J30,
60F17. Received: 25.01.2024
Accepted: 07.11.2024
DOI:
10.4213/tvp5703
Fulltext:
PDF file (696 kB)
First page:
PDF file