Time change equations for Lévy-type processes

We consider time change equations for Lévy-type processes. In this context we generalize the results of Böttcher et al. (2013) significantly. Namely, we are able to incorporate measurable instead of continuous multipliers. This opens a gate to find whole classes of symbols for which corresponding processes do exist. In order to establish our results we carefully analyze the connection between time change equations and classical initial value problems. This relationship allows us to transfer well-known results from this classical subject of pure mathematics into the theory of stochastic processes. On the way to prove our main theorem we establish generalizations of results on paths of Lévy-type processes.