The Annals of Applied Probability

Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process

Fabien Panloup

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We study some recursive procedures based on exact or approximate Euler schemes with decreasing step to compute the invariant measure of Lévy driven SDEs. We prove the convergence of these procedures toward the invariant measure under weak conditions on the moment of the Lévy process and on the mean-reverting of the dynamical system. We also show that an a.s. CLT for stable processes can be derived from our main results. Finally, we illustrate our results by several simulations.

Article information

Ann. Appl. Probab., Volume 18, Number 2 (2008), 379-426.

First available in Project Euclid: 20 March 2008

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Zentralblatt MATH identifier

Primary: 60H35: Computational methods for stochastic equations [See also 65C30] 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes
Secondary: 60F05: Central limit and other weak theorems

Stochastic differential equation Lévy process invariant distribution Euler scheme almost sure central limit theorem


Panloup, Fabien. Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process. Ann. Appl. Probab. 18 (2008), no. 2, 379--426. doi:10.1214/105051607000000285.

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  • Barndorff-Nielsen, O., Mikosh, T. and Resnick, S. (2001). Lévy Processes: Theory and Applications. Birkhäuser, Boston.
  • Berestycki, J. (2004). Exchangeable fragmentation-coalescence processes and their equilibrium measures. Electron J. Probab. 9 770–824.
  • Berkes, I., Horvath, L. and Khoshnevisan, D. (1998). Logarithmic averages of stable random variables are asymptotically normal. Stoch. Process. Appl. 77 35–51.
  • Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. Wiley, New York.
  • Brosamler, G. (1988). An almost everywhere central limit theorem. Math. Proc. Cambridge Phil. Soc. 104 561–574.
  • Ethier, S. and Kurtz, T. (1986). Markov Processes, Characterization and Convergence. Wiley, New York.
  • Deng, S. (2000). Pricing electricity derivatives under alternative stochastic spot price models. In Proceedings of the 33rd Hawaii International Conference on System Sciences 4. IEEE, Washington, DC.
  • Duflo, M. (1997). Random Iterative Models. Springer, Berlin.
  • Gnedenko, B. and Kolmogorov, A. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA.
  • Hall, P. and Heyde, C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Has’minskii, R. Z. (1981). Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen aan de Rijn.
  • Jacod, J. and Protter, P. (1991). Une remarque sur les équations différentielles à solutions Markoviennes. Séminaire de Probabilités XXV. Lecture Notes in Math. 1485 138–139. Springer, Berlin.
  • Lamberton, D. and Pagès, G. (2002). Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 367–405.
  • Lamberton, D. and Pagès, G. (2003). Recursive computation of the invariant distribution of a diffusion: The case of a weakly mean reverting drift. Stoch. Dynamics 4 435–451.
  • Lemaire, V. (2005). Estimation numérique de la mesure invariante d’une diffusion. Ph.D. thesis, Univ. de Marne-La Vallée. Available at
  • Lemaire, V. (2007). An adaptative scheme for the approximation of dissipative systems. Stochastic Process. Appl. 117 1491–1518.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Pagés, G. (2001). Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM Probab. Statist. 5 141–170.
  • Panloup, F. (2007). Computation of the invariant measure of a Lévy driven SDE: Rate of convergence. Stochastic Process. Appl. To appear.
  • Panloup, F. (2006). Approximation récursive du régime stationnaire d’une EDS avec sauts. Ph.D. thesis, Univ. de Paris VI. Available at
  • Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
  • Protter, P. and Talay, D. (1997). The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25 393–423.
  • Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales. 2. Wiley, New York.
  • Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • Schatte, P. (1988). On strong versions of the central limit theorem. Math. Nachr. 137 249–256.
  • Stout, W. F. (1979). Almost sure invariance principles when EX12=∞. Z. Wahrsch. Verw. Gebiete 49 23–32.
  • Talay, D. (1990). Second order discretization schemes of stochastic differential systems for the computation of the invariant law. Stoch. Stoch. Rep. 29 13–36.