Title: Estimating reducible stochastic differential equations by conversion to a least-squares problem,
Abstract: Stochastic differential equations (SDEs) are increasingly used in
longitudinal data analysis, compartmental models, growth modelling, and other
applications in a number of disciplines. Parameter estimation, however,
currently requires specialized software packages that can be difficult to use
and understand. This work develops and demonstrates an approach for estimating
reducible SDEs using standard nonlinear least squares or mixed-effects
software. Reducible SDEs are obtained through a change of variables in linear
SDEs, and are sufficiently flexible for modelling many situations. The approach
is based on extending a known technique that converts maximum likelihood
estimation for a Gaussian model with a nonlinear transformation of the
dependent variable into an equivalent least-squares problem. A similar idea can
be used for Bayesian maximum a posteriori estimation. It is shown how to obtain
parameter estimates for reducible SDEs containing both process and observation
noise, including hierarchical models with either fixed or random group
parameters. Code and examples in R are given. Univariate SDEs are discussed in
detail, with extensions to the multivariate case outlined more briefly. The use
of well tested and familiar standard software should make SDE modelling more
transparent and accessible. Keywords: stochastic processes; longitudinal data;
growth curves; compartmental models; mixed-effects; R