Ornstein–Uhlenbeck related models driven by Lévy processes

Recently, there has been increasing interest in continuous-time stochastic models with jumps, a class of models which has applications in the fields of finance, insurance mathematics and storage theory, to name just a few. In this chapter we shall collect known results about a prominent class of these continuoustime models with jumps, namely the class of Lévy-driven Ornstein–Uhlenbeck processes, and their generalisations. In Section 6.2, basic facts about Lévy processes, needed in the sequel, are reviewed. Then, in Section 6.3.1 the Lévydriven Ornstein–Uhlenbeck process, defined as a solution of the stochastic differential equation dVt = −λVtdt+ dLt, where L is a driving Lévy process, is introduced. An application to storage theory is mentioned, followed by the volatility model of Barndorff-Nielsen and Shephard (2001a, 2001b). Then, in Sections 6.3.2 and 6.3.3, two generalisations of Ornstein–Uhlenbeck processes are considered, both of which are based on the fact that an Ornstein–Uhlenbeck process can be seen as a continuoustime analogue of an AR(1) process with i.i.d. noise. In Section 6.3.2 we consider CARMA processes, which are continuous-time analogues of discrete time ARMA processes, and in Section 6.3.3 we consider generalised Ornstein– Uhlenbeck processes, which are continuous time analogues of AR(1) processes