, Volume 81, Issue 8, pp 953–983 | Cite as

Approximate maximum likelihood estimation for stochastic differential equations with random effects in the drift and the diffusion

  • Maud DelattreEmail author
  • Valentine Genon-Catalot
  • Catherine Larédo


Consider N independent stochastic processes \((X_i(t), t\in [0,T])\), \(i=1,\ldots , N\), defined by a stochastic differential equation with random effects where the drift term depends linearly on a random vector \(\Phi _i\) and the diffusion coefficient depends on another linear random effect \(\Psi _i\). For these effects, we consider a joint parametric distribution. We propose and study two approximate likelihoods for estimating the parameters of this joint distribution based on discrete observations of the processes on a fixed time interval. Consistent and \(\sqrt{N}\)-asymptotically Gaussian estimators are obtained when both the number of individuals and the number of observations per individual tend to infinity. The estimation methods are investigated on simulated data and show good performances.


Asymptotic properties Discrete observations Estimating equations Parametric inference Random effects models Stochastic differential equations 

Supplementary material

184_2018_666_MOESM1_ESM.pdf (232 kb)
Supplementary material 1 (pdf 231 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Maud Delattre
    • 1
    Email author
  • Valentine Genon-Catalot
    • 2
  • Catherine Larédo
    • 3
    • 4
  1. 1.UMR MIA-Paris, AgroParisTech, INRAUniversité Paris-SaclayParisFrance
  2. 2.UMR CNRS 8145, Laboratoire MAP5, Université Paris Descartes, Sorbonne Paris CitéParisFrance
  3. 3.INRA, MaIAGEJouy-en-JosasFrance
  4. 4.LPMA, Paris Diderot, Sorbonne Paris CitéParisFrance

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