# Simulation and approximation of Lévy-driven stochastic differential equations

ESAIM: Probability and Statistics (2012)

- Volume: 15, page 233-248
- ISSN: 1292-8100

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topFournier, Nicolas. "Simulation and approximation of Lévy-driven stochastic differential equations." ESAIM: Probability and Statistics 15 (2012): 233-248. <http://eudml.org/doc/222465>.

@article{Fournier2012,

abstract = {
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. Gebiete32 (1975) 111–131].
},

author = {Fournier, Nicolas},

journal = {ESAIM: Probability and Statistics},

keywords = {Lévy processes; stochastic differential equations;
Monte-Carlo methods; simulation; Wasserstein distance; Lévy process; stochastic differential equation; Monte Carlo method; Euler method},

language = {eng},

month = {1},

pages = {233-248},

publisher = {EDP Sciences},

title = {Simulation and approximation of Lévy-driven stochastic differential equations},

url = {http://eudml.org/doc/222465},

volume = {15},

year = {2012},

}

TY - JOUR

AU - Fournier, Nicolas

TI - Simulation and approximation of Lévy-driven stochastic differential equations

JO - ESAIM: Probability and Statistics

DA - 2012/1//

PB - EDP Sciences

VL - 15

SP - 233

EP - 248

AB -
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. Gebiete32 (1975) 111–131].

LA - eng

KW - Lévy processes; stochastic differential equations;
Monte-Carlo methods; simulation; Wasserstein distance; Lévy process; stochastic differential equation; Monte Carlo method; Euler method

UR - http://eudml.org/doc/222465

ER -

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