几类风险模型的首次通过时间及分红问题的研究
[Abstract]:In insurance mathematics, ruin theory is one of the most important issues in the study of insurance risk theory. It can provide a very useful early warning means for insurance company decision makers. Therefore, it is of great theoretical and practical significance to study it.
In this paper, the first passage time problem is considered for jump-diffusion processes by using the knowledge and theory of stochastic processes and stochastic differential equations. The ruin problem, the Gerber-Shiu function, and the problem of cash dividends and option pricing in the early stage of bankruptcy.
On the other hand, dividend strategy is also an important research issue in risk theory. "Dividend refers to the distribution of a company's (part) surplus as dividends to its owners or shareholders." Its practical significance makes the study of dividend strategy more concerned. For these beneficiaries, they are not only concerned about the company's current economic situation. What is more concerned about is what kind of dividend strategy can be adopted to make their earnings discounted at a certain discount rate as large as possible, that is, the so-called optimal dividend problem. According to different customer requirements, or under different dividend requirements, the optimal dividend strategy is naturally different. Now there are two commonly used strategies, one is the obstacle dividend. The other is the threshold dividend policy, which has been proved to be optimal under the corresponding restrictions. These two dividend strategies are discussed in Chapter 2, Chapter 3 and Chapter 5.
The first chapter mainly introduces the research background of this paper, including the basic risk model, dividend strategy, and the basic knowledge of L'evy process.
In Chapter 2, the first passage time of a hyperexponential jump (diffusion) process to a horizontal boundary is studied. The explicit solutions of the Laplace transformation are obtained for the first passage time, the first passage time and the distribution of overshoot, the process and the maximum (minimum) or the joint distribution. The compound Poisson risk model with diffusion disturbance and its dual model are given. The exact expression of the dividend formula under barrier dividend strategy and threshold dividend strategy is given.
Chapter 3 investigates the optimal dividend problem when the uncontrolled earnings process of an insurance company is a spectrum-negative L'evy process. Assuming that the dividend is distributed to customers in a constant proportion, the threshold strategy is proved to be the optimal dividend strategy when the L'evy measure has a completely monotone density. (The results of this chapter have been published in Acta Mathemat.) ICAE Applicatae Sinica, English Series.)
In Chapter 4, we study the first passage time of constant bounds for mixed exponential jump-diffusion processes. We obtain the explicit solution of Laplace transformation of the first passage time, the first passage time and the undershoot (overshoot) distribution or the joint distribution. We also obtain the explicit expression of Gerber-Shiu function for the two-sided jump-diffusion processes, and give the path. Analytical solutions of dependent options, Laplacian transformations of recall and barrier options, closed expressions of zero-interest loans with jumps in structured credit risk models.
In the fifth chapter, we study the optimal dividend problem of generalized composite Poisson model (whose counting process is a generalized Poisson process) and discuss its properties with classical risk model and Po'lya-Aeppli risk model as examples. The optimal dividend policy under certain conditions is the threshold strategy.
【学位授予单位】：曲阜师范大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F224;F840.31
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