Generalized Ornstein-Uhlenbeck Processes and Extensions

The generalized Ornstein-Uhlenbeck process Vt = e −ξt ( V0 + ∫ t 0 edηs ) , t ≥ 0, driven by a bivariate Lévy process (ξt, ηt)t≥0 with starting random variable V0 independent of (ξ, η) fulfills the stochastic differential equation dVt = Vt−dUt + dLt for another bivariate Lévy process (Ut, Lt)t≥0, which is determined completely by (ξ, η). In particular it holds ξt = − log(E(U)t), t ≥ 0, where E(U) denotes the stochastic exponential of U . In Chapter 2 of this work, for a given bivariate Lévy process (U, L), necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation dVt = Vt− dUt + dLt are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U, L) are assumed and noncausal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given. In Chapter 3 distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt−dUt+ dLt are analysed. In particular the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behaviour is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given. It is known that in many cases distributions of exponential integrals of Lévy processes, as they occur as stationary solutions of generalized Ornstein-Uhlenbeck processes, are infinitely divisible and in some cases they are also selfdecomposable. In Chapter 4, we give some sufficient conditions under which distributions of exponential integrals are not only selfdecomposable but furthermore are generalized gamma convolutions. We also study exponential integrals of more general independent increment processes. Several examples are given for illustration. Finally, in Chapter 5 a multivariate generalized Ornstein-Uhlenbeck process driven by a Lévy process (Xt, Yt)t≥0, with (Xt, Yt) ∈ R × R, d ≥ 1, is defined as Vt = E(X) t ( V0 + ∫ t 0 E(X)s−dYs ) , t ≥ 0.