Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient

Maud Delattre (1) , Valentine Genon-Catalot (2) , Adeline Samson (3)

1 MIA-Paris - Mathématiques et Informatique Appliquées
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 SAM - Statistique Apprentissage Machine

We consider $N$ independent stochastic processes $(X_i(t), t\in [0,T_i])$, $i=1,\ldots, N$, defined by a stochastic differential equation with diffusion coefficients depending on a random variable $\phi_i$. The distribution of the random effect $\phi_i$ depends on unknown population parameters which are to be estimated from a discrete observation of the processes $(X_i)$. The likelihood generally does not have any closed form expression. Two estimation methods are proposed: one based on the Euler approximation of the likelihood and another based on estimations of the random effects. When the distribution of the random effects is Gamma, the asymptotic properties of the estimators are derived when both $N$ and the number of observations per subject tend to infinity. The estimators are computed on simulated data for several models and show good performances.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Maud Delattre, Valentine Genon-Catalot, Adeline Samson. Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient. 2014. ⟨hal-00954438⟩