Paul Krühner: Time change equations for Lévy type processes

Pathwise transformations of stochastic processes have been first studied by Lamperti ('72). Their original use is to analyse a certain type of processes by transforming them into a hopefully simpler type of process. For instance, Lamperti ('79) intorduced the transformation to study positve self-similar processes by transforming them into L\'evy processes and Engelbert, Schmidt ('85) uses them to study weak solutions of the one-dimensional equation $dX(t) = \sigma(X(t)) dW(t)$ where $W$ is a standard Brownian motion which allow them to state exact existence and uniqueness results. Their result is based on detailed studies of the path behaviour of the Brownian motion. We generalise their approach to Lévy type processes and incorporate recent studies of Schnurr ('13).