ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat

Statistics of diffusion processes


The aim of this course is to present some techniques for estimating parameters present in a diffusion model. We treat the case where the trajectory is observed continuously and the case where the scattering is observed discretely with a $Delta>0$ time interval between each observation.


  1. Limit theorems for diffusion processes (ergodicity conditions, invariant measure, LLN and associated CLT)
  2. Estimation of a drift parameters for a continuously observed diffusion (study of likelihood, consistency of the corresponding estimator, examples)
  3. Estimation of the invariant measure
  4. Estimation of parameters from the discrete observation $(X_{iDelta})_{i=0,dots,n}$ of a diffusion : the cases of Black et Scholes and Ornstein Ulhenbeck processes.
  5. Estimation of volatility pararameters from high frequency discrete observations. Estimation functions and applications to diffusion models.
  6. Jump detection. "Multipower Realized Variation" estimators.


  • Richard Durett (1996), Stochastic calculus : a practical introduction (probability and stochastic series
  • Rafail Khasminskii (2012), Stochastic Stability of Differential Equations 2nd ed. Springer
  • Yury Kutoyants (2003) Statistical Inference for Ergodic Diffusion Processes, Springer Series in Statistics, Springer Verlag
  • Michael Sørensen (1998): Estimating functions for discretely observed diffusions: A review. In Basawa, I.V., Godambe, V.P. and Taylor, R.L. (eds.): Selected Proceedings of the Symposiumon Estimating Functions. IMS Lecture Notes - Monograph Series, Vol. 32, 305 - 325