Martingale-Transformations of Point-Processes and their Applications

Abstract

Non-negative martingales represent an important aspect of process the- ory. The main contribution of this thesis is to present methods to transform a general point process N, characterized by its conditional intensity λ, into a process that satisfies the properties of a non-negative martingale. The resulting systems allow the development of techniques for the investigation of statistical problems. Historically, martingale transformations have been studied for the first time with respect to Ito processes. In this case, where the triggering process is a Brownian motion, a backward heat equation must be solved. This theory has evolved successfully over several decades and can now be formulated in a generalized form for semimartingales. In the present work, these results are used, among other things, to generalize martingales that are already known in connection with the empirical distribution function. Within this approach, various solutions arise, whose associated processes exhibit different characteristics. This allows us to have a natural access to the class of Poisson-Charlier functions as a solution to the differential equation for the case of a Poisson process. The Poisson-Charlier functions are further generalized in a subsequent step for more general point processes. To generate non-negative martingales, we also make use of the integral representation for martingales by imposing characteristic properties on the generating predictable processes.