Estimation for stochastic differential equations with mixed effects.

Valentine Genon-Catalot 1 Catherine Larédo 2
2 M.I.A., I.N.R.A.
LPMA - Laboratoire de Probabilités et Modèles Aléatoires, MIA - Unité de recherche Mathématiques et Informatique Appliquées
Abstract : We consider the long term behaviour of a one-dimensional mixed effects diffusion process $(X(t))$ with a multivariate random effect $\phi$ in the drift coefficient. We first study the estimation of the random variable $\phi$ based on the observation of one sample path on the time interval $[0,T]$ as $T$ tends to infinity. The process $(X(t))$ is not Markov and we characterize its invariant distributions. We build moment and maximum likelihood-type estimators of the random variable $\phi$ which are consistent and asymptotically mixed normal with rate $\sqrt{T}$. Moreover, we obtain non asymptotic bounds for the moments of these estimators. Examples with a bivariate random effect are detailed. Afterwards, the estimation of parameters in the distribution of the random effect from $N$ {\em i.i.d.} processes $(X_j(t), t\in [ 0, T]), j=1,\ldots,N$ is investigated. Estimators are built and studied as both $N$ and $T=T(N)$ tend to infinity. We prove that the convergence rate of estimators differs when deterministic components are present in the random effects. For true random effects, the rate of convergence is $\sqrt{N}$ whereas for deterministic components, the rate is $\sqrt{NT}$. Illustrative examples are given.
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Valentine Genon-Catalot, Catherine Larédo. Estimation for stochastic differential equations with mixed effects.. Statistics, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2016, 50 (5), pp.1014-1035. ⟨hal-00807258v3⟩

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