Markovianity and Time Changed Lévy Processes

The objective of the thesis is to investigate Markov structures of time changed Lévy processes. Moreover we investigate the Markov structures of stochastic differential equations driven by Brownian motion with a view towards obtaining Markov structures of stochastic differential equations driven by time changed Brownian motion. The bulk of the thesis consists of four chapters. A chapter for preliminary material, and a chapter for each of the three themes, Markovianity, Lévy processes and time changed Lévy processes. The chapter on Markovianity deals with different equivalent definitions of the Markov property and states results on the finite dimensional distribution of a Markov process. The chapter on Lévy processes deals with the definitions of Lévy processes, the Lévy-Khintchine representation, and makes the connections between Lévy processes and Markov processes, showing that a Lévy process is a Markov process. The chapter on time changed processes deals, at first, with general time changed progressively meausurable stochatic processes. Here we are interested in showing that such processes are measurable. We present the well know result that a subordination process is a Markov process. Moreover we characterize time changed Lévy processes by the property of conditionally stationary independent increments. In this chapter we deal with two well-known time changed Lévy processes, namely the Normal inverse Gaussian distribution and the Cox process, also known as a doubly stochastic Poisson process. Moreover measuraility properties of the time changed Lévy process is studied, leading to different types of filtrations with respect to which the time changed Lévy process is measurable. With one of these fitrations it is possible to show a result, that is interesting in the study of Markovianity of time changed Lévy processes. Throughout the chapters we unfold results on the Markovianity of solutions to stochastic differential equations and present a result that shows that a solution to a stochastic differential equation driven by a Brownian motion time changed with a subordinator (the time process is a Lévy process) can be shown to exist and can be shown to be a Markov process. Moreover we introduce the case where the time change process is continuous