Analysis Of Some Jump-diffusion Process And Their Application In Risk Theory

Collective risk theory, considering a model of the risk business of an insurance company, has been studied for more than a century and always been a vital part of actuarial mathematics. It is well known that the classical risk model was firstly introduced by Lundberg (1903), and was later developed and made mathematically rigorous by Harald Cramer in Cramer (1930). This model, also known as the Cramér-Lundberg risk model, models the surplus of an insurance business as a compound Poisson process and has since then been studied and extended in various ways. The perturbed compound Poisson risk process of Gerber (1970) and the Sparre Andersen risk process of Andersen (1957) are the most famous extensions. Together with the generalizations of the model, there are diversified approaches and functionals introduced to study risk process. Renewal theory, Wiener-Hoff equation, It(o|^) formula, piecewise deterministic Markov processes and the martingale methods are some of the most commonly used approaches in risk theory and other disciplines. See Rolski et al. (1999) and references therein. And three most important actuarial variables, the time of ruin, the deficit at ruin, the surplus immediately before ruin, are also captured by introduction of a so-called Gerber-Shiu discounted penalty function in Gerber and Shiu (1997, 1998a) and Tsai and Willmot (2002).On the other hand, due to its practical importance, the issue of dividend strategies becomes an increasingly popular branch of risk theory. Dividends are payments made to stockholders from a firm's earnings, it is desirable to find a fixed rule which produces largest possible expected sum of discounted dividend, and that is the optimal dividend problem. Among all the dividend policies associated with different criterions, the barrier dividend strategy and the threshold dividend strategy are of particular interest, and in the literature of classical risk model, they are the "best" under their corresponding constraints. Therefore, we consider risk models with the presence of these dividend strategies in Chapter 2 and Chapter 3. And more recently, there is another hot topic of so-called multi-layer threshold dividend model which becomes popular, a Markov-modulated risk model with this dividend strategy is studied in Chapter 4. On the basis of these background, this thesis is mainly devote to the analysis of the Gerber-Shiu discounted penalty function(also called Gerber-Shiu function later) and the expected discounted dividend function of some risk model.Firstly, we shortly review the history of risk theory, the most commonly used methods, functions and models are presented in the first Chapter, as well as those now classical works. And the organization of this thesis is also given in this chapter. Then, the main body of the thesis starts.The next two chapters are mainly concerned on the proof of differentiable of the functions at the dividend barriers. In chapter 2, we considered the perturbed compound Poisson risk model with a threshold dividend strategy. The expected discounted dividend function and the Gerber-Shiu discounted penalty function were studied. Inspired by Wang and Wu (2000) and Wang (2001), we proved the boundary conditions, following the common way of solving integro-differential equations, and expressed the function concerned in terms of some other functions.In chapter 3, motivated by the model in Avanzi et al. (2007), we considered a dual model with perturbations of Brownian motion and barrier dividend. The expected discounted dividend function was studied. And we also aimed at finding its boundary conditions, and then expressed it in the usual way.And in chapter 4, a perturbed renewal jump-diffusion risk model is studied, the conclusions are extended to a Markov-modulated model with the claim sizes specified to be in the class of phase type distributions. Motivated by Asmussen (1995) and Bladt (2005), following the fluid flow method, we studied some passage time of the process, and expressed the ruin related quantities in matrix form, such as the joint distribution of the ruin time, the surplus before ruin and the deficit at ruin. And we also success in providing the necessary steps that should be taken to ensure the solvability of matrix equation considered, as well as the uniqueness of the solution. And then, by applying the conclusions from previous chapters, we also studied this process with perturbation of threshold dividend and multi-layer dividend strategy.