Occupation time problems in some insurance risk models

© 2016 Dr Can JinThis thesis studies occupation times and related quantities through their Laplace transforms by adopting two diff erent approaches in various risk models. The occupation time in a risk model refers to the total time that the underlying risk process spends while it stays within a particular interval prior to a stopping time. In this thesis we choose the stopping time to be the time of ruin, i.e. the first time that the risk process falls below zero. The risk models considered in this thesis include the classical risk model, the Markovian arrival risk model and the Sparre Andersen risk model in both a continuous and a discrete setting with the possibility of a deferral. The first approach builds on the usage of system of (integro)-differential equations which are traditionally used in both applied probability and risk theory literature to study fi rst passage time problems. The Laplace transforms of the occupation times satisfy certain (integro)-diff erential equations, which can be solved to yield explicit expressions for the underlying Laplace transforms. Together with probabilistic arguments, we obtain explicit expressions for the moments of the occupation times which are easily computable. In addition, Laplace inversion techniques allow us to obtain analytical expressions for the density functions which help us to study the distributional properties of the occupation times The other approach builds on the theory of fluctuation techniques which involve the so-called scale functions de fined for L evy processes. Thanks to the independent increments property of the class of L evy processes, the Laplace transforms of the occupation times can be expressed in terms of scale functions without the original complexity when studying certain scale functions. In this thesis, one occupation time, which we name as the non-dividend paying duration, is fi rstly defi ned and studied in the literature via the use of scale functions. We remark that this particular quantity serves as an alternative tool in risk management when dividend arrangements are involved. Finally, we extend the analysis on the ruin related quantities into a discrete-time Sparre Andersen risk model with general inter-claim times and phase-type claim sizes. A generalized Gerber-Shiu function is studied and the joint distribution functions of the ruin related quantities are derived when the claim sizes follow a zero-truncated geometric distribution