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].