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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/ps/2009017/
[1] and , Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38 (2001) 482-493. | MR | Zbl
[2] , and , A method for simulating stable random variables. J. Amer. Statist. Assoc. 71 (1976) 340-344. | MR | Zbl
[3] , Extensions of results of Komlos, Major, and Tusnady to the multivariate case. J. Multivariate Anal. 28 (1989) 20-68. | MR | Zbl
[4] , Solving Landau equation for some soft potentials through a probabilistic approach. Ann. Appl. Probab. 13 (2003) 515-539. | MR | Zbl
[5] and , Stochastic differential equations and diffusion processes, Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo (1989). | MR | Zbl
[6] , The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Probab. 32 (2004) 1830-1872. | MR | Zbl
[7] , and , On asymptotic errors in discretization of processes. Ann. Probab. 31 (2003) 592-608. | MR | Zbl
[8] , , and , The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 523-558. | EuDML | Numdam | MR | Zbl
[9] and , Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 (1998) 267-307. | MR | Zbl
[10] and , Limit theorems for stochastic processes, second edition. Springer-Verlag, Berlin (2003). | MR | Zbl
[11] , and , An approximation of partial sums of independent rvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111-131. | MR | Zbl
[12] and , The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25 (1997) 393-423. | MR | Zbl
[13] , Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 802-817. | EuDML | Numdam | MR | Zbl
[14] , Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stochastic Process. Appl. 103 (2003) 311-349. | MR | Zbl
[15] and , Improved convergence rate for the simulation of stochastic differential equations driven by subordinated Lévy processes. Stochastic Process. Appl. 108 (2003) 1-26. | MR | Zbl
[16] , Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 (1978/79) 67-105. | MR | Zbl
[17] , On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rat. Mech. Anal. 143 (1998) 273-307. | MR | Zbl
[18] , A stochastic model of neural response. Adv. Appl. Prob. 13 (1981) 231-281. | MR | Zbl
[19] , Estimates for the strong approximation in multidimensional central limit theorem. Proceedings of the International Congress of Mathematicians, Vol. III (2002) 107-116. | MR | Zbl
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