The Annals of Applied Probability

Central limit theorem for the multilevel Monte Carlo Euler method

Mohamed Ben Alaya and Ahmed Kebaier

Full-text: Open access


This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607–617] which is significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg–Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [Ann. Probab. 26 (1998) 267–307], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. A complexity of the multilevel Monte Carlo algorithm is carried out.

Article information

Ann. Appl. Probab., Volume 25, Number 1 (2015), 211-234.

First available in Project Euclid: 16 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62F12: Asymptotic properties of estimators 65C05: Monte Carlo methods 60H35: Computational methods for stochastic equations [See also 65C30]

Central limit theorem multilevel Monte Carlo methods Euler scheme finance


Ben Alaya, Mohamed; Kebaier, Ahmed. Central limit theorem for the multilevel Monte Carlo Euler method. Ann. Appl. Probab. 25 (2015), no. 1, 211--234. doi:10.1214/13-AAP993.

Export citation


  • [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [2] Bouleau, N. and Lépingle, D. (1994). Numerical Methods for Stochastic Processes. Wiley, New York.
  • [3] Creutzig, J., Dereich, S., Müller-Gronbach, T. and Ritter, K. (2009). Infinite-dimensional quadrature and approximation of distributions. Found. Comput. Math. 9 391–429.
  • [4] Dereich, S. (2011). Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann. Appl. Probab. 21 283–311.
  • [5] Duffie, D. and Glynn, P. (1995). Efficient Monte Carlo simulation of security prices. Ann. Appl. Probab. 5 897–905.
  • [6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [7] Giles, M. (2008). Improved multilevel Monte Carlo convergence using the Milstein scheme. In Monte Carlo and Quasi-Monte Carlo Methods 2006 343–358. Springer, Berlin.
  • [8] Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 607–617.
  • [9] Giles, M. B., Higham, D. J. and Mao, X. (2009). Analysing multi-level Monte Carlo for options with nonglobally Lipschitz payoff. Finance Stoch. 13 403–413.
  • [10] Giles, M. B. and Szpruch, L. (2013). Multilevel Monte Carlo Methods for Applications in Finance. World Scientific, Singapore.
  • [11] Heinrich, S. (1998). Monte Carlo complexity of global solution of integral equations. J. Complexity 14 151–175.
  • [12] Heinrich, S. (2001). Multilevel Monte Carlo Methods. Lecture Notes in Computer Science 2179 58–67. Springer, Berlin.
  • [13] Heinrich, S. and Sindambiwe, E. (1999). Monte Carlo complexity of parametric integration. J. Complexity 15 317–341.
  • [14] Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2013). Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Ann. Appl. Probab. 23 1913–1966.
  • [15] Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 232–246. Springer, Berlin.
  • [16] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
  • [17] Kebaier, A. (2005). Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing. Ann. Appl. Probab. 15 2681–2705.
  • [18] Kloeden, P. E. and Platen, E. (1995). Numerical methods for stochastic differential equations. In Nonlinear Dynamics and Stochastic Mechanics 437–461. CRC Press, Boca Raton, FL.
  • [19] Korn, R., Korn, E. and Kroisandt, G. (2010). Monte Carlo Methods and Models in Finance and Insurance. CRC Press, Boca Raton, FL.
  • [20] Protter, P. (1990). Stochastic Integration and Differential Equations. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [21] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509 (1991).