A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

Change of measure up to fixed times or stopping times is the theme of the famous Cameron–Martin–Girsanov formula. The paper studies change of measure up to random times which are not stopping times of the natural filtration. The ultimate aim is to build up a family of interesting models for physics and chemistry.