Study On Ruin Problems In The Risk Models With Random Incomes

In classical risk models and many extended risk models, premiums are assumed to be received at a constant rate over time. However, this assumption is often too restrictive in applications. In order to give a reasonably realistic description of the actual behavior of the risk movement, many kinds of extended risk models are introduced in the field of risk theory. In this paper, we analyze the Gerber-Shiu discounted penalty function by using stochastic process, integro-differential equation theory for three risk models with random incomes. This thesis is organized as follows.In Chapter 1, we simply introduce the background of the risk theory and its development. And then several well-known properties and theorems are introduced in the preliminaries. Finally we present the main content of this thesis and the main result of my research.In Chapter 2, we consider a generalization of the classical ruin model, where the income is random and the distribution of the time between two claim occurrences depends on the previous claim size. This dependency was proposed in Albrecher and Boxma(2004) firstly. Explicit expressions for the generating function of the Gerber-Shiu penalty function are derived. Two similar models are also discussed. Finally, the result are shown by two examples. We further generalize the premium income into a compound Poisson process in Chapter 3. Given the premium size is exponentially distributed, the Gerber-Shiu penalty functions are derived.In Chapter 4, The main focus is to analyze the Gerber–Shiu penalty function of a compound Poisson risk model with delayed claims and random incomes. It is assumed that every main claim will produce a by-claim which can be delayed with a certain probability. We derive the integral equation satisfied by the Gerber–Shiu penalty function. Given that the premium size is exponentially distributed and Erlang(n,β) distributed, the explicit expressions for the Laplace transform of the Gerber–Shiu penalty functions are derived. Finally, when the premium sizes have rational Laplace transforms, we also obtain the Laplace transform of the Gerber–Shiu penalty function.