Ornstein–Uhlenbeck Processes and Extensions

This paper surveys a class of Generalised Ornstein-Uhlenbeck (GOU) processes associated with Lévy processes, which has been recently much analysed in view of its applications in the financial modelling area, among others. We motivate the Lévy GOU by reviewing the framework already well understood for the “ordinary” (Gaussian) Ornstein-Uhlenbeck process, driven by Brownian motion; thus, defining it in terms of a stochastic differential equation (SDE), as the solution of this SDE, or as a time changed Brownian motion. Each of these approaches has an analogue for the GOU. Only the second approach, where the process is defined in terms of a stochastic integral, has been at all closely studied, and we take this as our definition of the GOU (see Eq. (12) below). The stationarity of the GOU, thus defined, is related to the convergence of a class of “Lévy integrals”, which we also briefly review. The statistical properties of processes related to or derived from the GOU are also currently of great interest, and we mention some of the research in this area. In practise, we can only observe a discrete sample over a finite time interval, and we devote some attention to the associated issues, touching briefly on such topics as an autoregressive representation connected with a discretely sampled GOU, discrete-time perpetuities, self-decomposability, self-similarity, and the Lamperti transform. Some new statistical methodology, derived from a discrete approximation procedure, is applied to a set of financial data, to illustrate the possibilities.