Existence and uniqueness of stationary Lévy-driven CARMA processes

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining a general Lévy-driven continuous-parameter ARMA process with index set R are determined. Under these conditions the solution is shown to be unique and an explicit expression is given for the process as an integral with respect to the background driving Lévy process. The results generalize results obtained earlier for second-order processes and for processes defined by the Ornstein–Uhlenbeck equation.