The Annals of Probability

Asymptotic error distributions for the Euler method for stochastic differential equations

Jean Jacod and Philip Protter

Full-text: Open access


We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Itô’s equations the rate is $1/ \sqrt{n}$; we provide a necessary and sufficient condition for this rate to be $1/ \sqrt{n}$ when the driving semimartingale is a continuous martingale, or a continuous semimartingale under a mild additional assumption; we also prove that in these cases the normalized error processes converge in law.

The rate can also differ from $1/ \sqrt{n}$: this is the case for instance if the driving process is deterministic, or if it is a Lévy process without a Brownian component. It is again $1/ \sqrt{n}$ when the driving process is Lévy with a nonvanishing Brownian component, but then the normalized error processes converge in law in the finite-dimensional sense only, while the discretized normalized error processes converge in law in the Skorohod sense, and the limit is given an explicit form.

Article information

Ann. Probab., Volume 26, Number 1 (1998), 267-307.

First available in Project Euclid: 31 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 65U05
Secondary: 60G44: Martingales with continuous parameter 60F17: Functional limit theorems; invariance principles

Stochastic differential equations Euler scheme error distributions Lévy processes numerical approximation


Jacod, Jean; Protter, Philip. Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 (1998), no. 1, 267--307. doi:10.1214/aop/1022855419.

Export citation


  • [1] Aldous, D. J. and Eagleson, G. K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325-331.
  • [2] Delattre, S. and Jacod, J. (1996). A central limit theorem for normalized functions of the increments of a diffusion process, in presence of round-off errors. Bernoulli. To appear.
  • [3] Jacod, J. (1996). On continuous conditional Gaussian martingales and stable convergence in law. Publ. Lab. Probabilit´es 339, Univ. Paris-6.
  • [4] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorem for Stochastic Processes. Springer, Berlin.
  • [5] Jakubowski, A. (1995). A non-Skorohod topology on the Skorohod space. Preprint, Torun Univ.
  • [6] Jakubowski, A., M´emin, J. and Pag es, G. (1989). Convergence en loi des suites d'int´egrales stochastiques sur l'espace D1 de Skorohod. Probab. Theory Related Fields 81 111-137.
  • [7] Kurtz, T. G. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070.
  • [8] Kurtz, T. G. and Protter, P. (1991). Wong-Zakai corrections, random evolutions and numerical schemes for SDEs. In Stochastic Analysis 331-346. Academic Press, New York.
  • [9] Kurtz, T. G. and Protter, P. (1996). Weak convergence of stochastic integrals and differential equations. 1995 CIME School in Probability. Lecture Notes in Math. 1627 1-41. Springer, Berlin.
  • [10] M´emin, J. and Slomi ´nski, L. (1991). Condition UT et stabilit´e en loi des solutions d'´equations diff´erentielles stochastiques. S´eminaire de Probabilit´es XXV. Lecture Notes in Math. 1485 162-177. Springer, Berlin.
  • [11] Meyer, P. A. and Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincar´e Probab. Statist. 20 353-372.
  • [12] Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
  • [13] R´enyi, A. (1963). On stable sequences of events. Sanky¯a Ser. A 25 293-302.
  • [14] Slomi ´nski, L. (1989). Stability of strong solutions of stochastic differential equations. Stochastic Process. Appl. 31 173-202.
  • [15] Slomi ´nski, L. (1996). Stability of stochastic differential equations driven by general semimartingales. Dissertationes Math. 349 1-113.
  • [16] Talay, D. (1995). Simulation of stochastic differential systems. Probabilistic Methods in Applied Physics. Lecture Notes in Physics 451 63-106. Springer, New York.
  • [17] Tukey, J. W. (1939). On the distribution of the fractional part of a statistical variable. Mat Sb. N.S. 4 561-562.