ABSTRACT
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 Le´vy-driven Ornstein-Uhlenbeck processes, and their generalisations. In Section 6.2, basic facts about Le´vy processes, needed in the sequel, are reviewed. Then, in Section 6.3.1 the Le´vydriven Ornstein-Uhlenbeck process, defined as a solution of the stochastic differential equation
dVt = −λVtdt+ dLt, where L is a driving Le´vy 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 OrnsteinUhlenbeck processes, which are continuous time analogues of AR(1) processes
with i.i.d. random coefficients. Special emphasis is given to conditions for stationarity of these processes and existence of moments. Then, in Section 6.3.4, we introduce the COGARCH(1,1) process, which is a continuous-time analogue of the ARCH process of Engle (1982) and the GARCH(1,1) process of Bollerslev (1986). It is an example of a generalised Ornstein-Uhlenbeck process. An extension to COGARCH(q, p) processes is also given.
