PhD Defense Phillip Frede Halmsted Olsen

Title: Matrix methods for Markovian and semi-Markovian processes

Abstract:
This thesis addresses a selection of theoretical and applied mathematical problems using matrix-based methods which are rooted in applied probability. We first propose a likelihood-based approach to obtaining phase-type jump diffusion approximations of generalised hyperexponential Lévy processes, and study its numerical application in several examples. Then we define the class of Bernstein-induced matrix distributions, which are positive distributions whose Laplace transforms are rational functions composed with Bernstein functions. We show that this construction leads to a generalisation of the class of phase-type distributions, which allows for more versatile tail behaviour while retaining useful closure properties and probabilistic sample path characterisations. Next we study a time-inhomogeneous extension of the Markovian arrival process with marked arrivals and batch arrivals. We derive distributional properties and provide estimation procedures for a dense subclass in terms of two EM algorithms. Finally, we introduce a new matrix based framework for modelling time-inhomogeneous semi-Markov processes which is based on extensions of the product integral. We apply these tools to the analysis of reward processes, where we establish explicit matrix representations for higher order moments and transforms for discounted accumulated rewards

Online link: To be announced soon

Thesis  To be announced soon

Principal supervisor:
Professor Mogens Bladt, University of Copenhagen
Co-supervisor: Professor emeritus Thomas Mikosch, University of Copenhagen

Assessment committee:
Professor Jeffrey F. Collamore (Chairman), University of Copenhagen
Professor Marcus C. Christiansen, Carl von Ossietzky Universität Oldenburg
Professor David Landriault University of Waterloo