  Continuous-time state-space models (SSMs) are flexible tools for analysing
irregularly sampled sequential observations that are driven by an underlying
state process. Corresponding applications typically involve restrictive
assumptions concerning linearity and Gaussianity to facilitate inference on the
model parameters via the Kalman filter. In this contribution, we provide a
general continuous-time SSM framework, allowing both the observation and the
state process to be non-linear and non-Gaussian. Statistical inference is
carried out by maximum approximate likelihood estimation, where multiple
numerical integration within the likelihood evaluation is performed via a fine
discretisation of the state process. The corresponding reframing of the SSM as
a continuous-time hidden Markov model, with structured state transitions,
enables us to apply the associated efficient algorithms for parameter
estimation and state decoding. We illustrate the modelling approach in a case
study using data from a longitudinal study on delinquent behaviour of
adolescents in Germany, revealing temporal persistence in the deviation of an
individual's delinquency level from the population mean.
