We consider the approximate Euler scheme for Lévy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Lévy measure of the driving process behaves like |z|-1-αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the small jumps, see [S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stochastic Process. Appl. 103 (2003) 311-349], the computational cost is of order nα/(2-α), which is huge when α is close to 2. In the same spirit, we study the problem of the approximation of a Lévy-driven S.D.E. by a Brownian S.D.E. when the Lévy process has no large jumps. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 802-817] about the central limit theorem, in the spirit of the famous paper by Komlós-Major-Tsunády [J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independent rvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111-131].
Mots clés : Lévy processes, stochastic differential equations, Monte-Carlo methods, simulation, Wasserstein distance
@article{PS_2011__15__233_0,
author = {Fournier, Nicolas},
title = {Simulation and approximation of {L\'evy-driven} stochastic differential equations},
journal = {ESAIM: Probability and Statistics},
pages = {233--248},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2009017},
mrnumber = {2870514},
zbl = {1273.60080},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2009017/}
}
TY - JOUR AU - Fournier, Nicolas TI - Simulation and approximation of Lévy-driven stochastic differential equations JO - ESAIM: Probability and Statistics PY - 2011 SP - 233 EP - 248 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2009017/ DO - 10.1051/ps/2009017 LA - en ID - PS_2011__15__233_0 ER -
%0 Journal Article %A Fournier, Nicolas %T Simulation and approximation of Lévy-driven stochastic differential equations %J ESAIM: Probability and Statistics %D 2011 %P 233-248 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2009017/ %R 10.1051/ps/2009017 %G en %F PS_2011__15__233_0
Fournier, Nicolas. Simulation and approximation of Lévy-driven stochastic differential equations. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 233-248. doi : 10.1051/ps/2009017. http://www.numdam.org/articles/10.1051/ps/2009017/
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