Time Change Equations for L\'evy Type Processes

In this paper we analyse time change equations (TCEs) for L\'evy-type processes in detail. To this end we establish a connection between TCEs and classical one-dimensional initial value problems (IVPs) which are easier to handle. Properties of the IVPs are linked with properties of the TCEs. We show in a general setting existence and uniqueness of solutions of the TCEs. Our main result is based on the general path properties for L\'evy-type processes found in Schnurr (2013). Applications include an existence result for processes which correspond to a certain class of given symbols.