Hitting times in urn models and occupation times in one-dimensional diffusion models

The main subject of this thesis is certain functionals of Markov processes. The thesis can be said to consist of three parts. The first part concerns hitting times in urn models, which are Markov processes in discrete time. In particular, the expected time to absorption in the Mabinogion model is studied. For instance, we give formulas for the expected time to absorption as a function of the initial state of the process, both in the ordinary Mabinogion model and under a strategy that solves an optimal control problem. The second part of the thesis is about occupation times of one-dimensional diffusions, which are continuous Markov processes. We give a recursive formula for the moments of the occupation time on the positive real line, in the case that the diffusion has a self-similar property, or for the Laplace transform of the moments, in case of a general diffusion. The recurrence is based on the Green kernel of the diffusion. In the third part of the thesis, we give results on some combinatorial summation identities that are connected to the other presented results. These include double sums with ratios of binomial coefficients, as well as sums involving Stirling numbers of both first and second kind.