Long-time behavior of markov additive processes

MAPs, which were introduced in the early 1970s, have been studied and widely used to model systems subject to regime-switching behavior, such as portfolio optimization, inventory management and reliability engineering. In these models, the underlying Markov chain on a finite state space serves as a modulator that tracks the current regime or state of the system. The aim of the thesis is to study general Markov additive processes when the state space of the modulator is a Polish space. In the first part, we give a characterization of the long-time behavior of the ordinate in terms of the distributions of the last time at maximum and the last time at minimum under some regularity assumptions. Then, we give this characterization in terms of the associated ladder time process and the excursion measure under the reversibility property assumption. We extend this result under the weak-reversibility property assumption, which is weaker than the reversibility property. In the second part of the thesis, we show the applicability of our assumptions on some well-known self-similar Markov processes. For this purpose, we consider the Lamperti-Kiu transform, which gives a correspondence between d-dimensional self-similar Markov processes and Markov additive processes which the state space of the modulator is in (d-1)-dimensional unit sphere. The asymptotic behavior of the radial distance from the origin of a self-similar Markov process can be characterized by the long-time behavior of the ordinate of the corresponding Markov additive process. We work on two important classes of self-similar Markov processes and show that the corresponding Markov additive processes satisfy the main assumptions. In the third part of the thesis, we give the chaotic and predictable representation theorems for Markov additive processes. We consider the case where the modulator is a real-valued Lévy process with bounded variation. We construct the necessary extra power jumps processes corresponding to the different parts of the Markov additive processes. The chaotic representation of square-integrable random variables is given in terms of certain pairwise strongly orthonormal martingales constructed from the additional processes. Then, the predictable representation of square-integrable martingales is given in terms of the ordinate and the power jump processes.