Time-changed SDEs and connection to PDEs

In the classical theory of stochastic processes, there is a relationship between the stochastic process, the stochastic differential equation (SDE) driven by it, and the Fokker-Planck-Kolmogorov (FKP) equations associated with them. That relationship is well documented for the triad of Brownian motion, Ito SDEs driven by Brownian motion and their associated FPK equations. In this work, we discuss a possible generalization of that relationship. We introduce the stochastic calculus where the driving processes are semimartingales and their time changed versions. A special version of stochastic calculus with SDEs driven by time changed martingales is then developed. Exploring the relationship between SDEs driven by time changed Levy process and its associated FPK equation, it is found that the latter is given by the time-fractional differential operators. Lastly, a heuristic approach to a CTRW approximation having a time changed Levy process as a limit is given, alongside with the associated PDE of the limit.