Parameter Estimation in Stochastic Differential Mixed-Effects Models

Stochastic differential equation (SDE) models have shown useful to describe continuous time processes, e.g. a physiological process evolving in an individual. Biomedical experiments often imply repeated measurements on a series of individuals or experimental units and individual differences can be represented by incorporating random effects into the model. When both system noise and individual differences are considered, stochastic differential mixed effects models ensue. In most cases the likelihood function is not available, and thus maximum likelihood estimation is not possible. Here we propose to approximate the unknown likelihood function by first approximating the conditional transition density of the diffusion process given the random effects by a Hermite expansion, as suggested by Aït-Sahalia (2001, 2002), and then numerically integrate the obtained conditional likelihood with respect to the random effects. The approximated maximum likelihood estimators are evaluated on simulations from the Ornstein-Uhlenbeck process and Geometric Brownian motion.