• biomedical applications;
  • Brownian motion with drift;
  • CIR process;
  • closed-form transition density expansion;
  • Gaussian quadrature;
  • geometric Brownian motion;
  • maximum likelihood estimation;
  • Ornstein–Uhlenbeck process;
  • random parameters;
  • stochastic differential equations

Abstract.  Stochastic differential equations have been shown useful in describing random continuous time processes. Biomedical experiments often imply repeated measurements on a series of experimental units and differences between units can be represented by incorporating random effects into the model. When both system noise and random effects are considered, stochastic differential mixed-effects models ensue. This class of models enables the simultaneous representation of randomness in the dynamics of the phenomena being considered and variability between experimental units, thus providing a powerful modelling tool with immediate applications in biomedicine and pharmacokinetic/pharmacodynamic studies. In most cases the likelihood function is not available, and thus maximum likelihood estimation of the unknown parameters is not possible. Here we propose a computationally fast approximated maximum likelihood procedure for the estimation of the non-random parameters and the random effects. The method is evaluated on simulations from some famous diffusion processes and on real data sets.