Exit problems for fractional processes, random walks and an insurance model

Abstract

In this thesis, we are concerned with different persistence problems and a problem from risk theory. The content of this work is divided into four parts that are mostly independent. In the first part, we consider two classes of Gaussian sequences that are discrete-time analogs of two-sided fractional Brownian motion and two-sided integrated fractional Brownian motion, respectively. In both cases, we show that the persistence probability decreases polynomially and determine the polynomial rate. In the second part, we present a first contribution to the rigorous study of fractional Brownian motion conditioned to be positive. More precisely, we consider a slightly modified problem, where the process is penalized instead of being killed when becoming negative. Then, we discuss the result in the Brownian case in terms of stochastic differential equations. In the third part, we generalize classical persistence questions for centered random walks with finite variance. For this purpose, we introduce a class of absorption mechanisms that generalize the classical situation. Our main results serve as a toolkit that allows to obtain persistence probability and scaling limit results for many different examples in this class. In the fourth part, we are concerned with a problem from risk theory. We consider the classical Cramer-Lundberg process but with modified notion of ruin. Then, under a rather general assumption on our model, which is satisfied by most of such modified models from the literature, we study the relation of the asymptotics of the modified ruin probability to the classical ruin probability. This is done under the Cramer condition as well as for subexponential integrated claim sizes.