Study On Ruin Problems Of The Discrete Time Semi-markov Risk Models

Research on related problems of insurance has been a hot topic in the risk theory.Compared with the classical financial insurance risk model,the Markov-modulated risk models are more consistent with the financial and insurance data in reality.By constructing an external Markov environment,the semi-Markov risk model determined by the state transition describes the changes for the insurance company and the influence of various factors on the actual operation.The semi-Markov risk model is available to handle the dependencies arisen in the operation of insurance company.On the other hand,the conditional independence properties make the model more convenient in mathematics.Therefore,discrete time semi-Markov risk model has theoretical and practical value.This paper mainly studies three kinds of risk models based on discrete time semi-Markov process.By using the technique of probability generating function,the recursive formulae of ruin probability or Gerber-Shiu expected discounted penalty function are given.The theoretical results are analyzed by a numerical example.This thesis is divided into four chapters.In chapter 1,the development process and theoretical value of insurance theory are introduced firstly.Then the preliminary knowledge related to discrete time semi-Markov risk model is given,such as the definition and properties of Markov process,semi-Markov process,probability generation function and Gerber-Shiu expected discounted penalty function.In chapter 2,this chapter focuses on the discrete time semi-Markov risk model to study the survival probability of the model when the premium is random.With the technique of probability generating function,we get the recursive formula of the survival probability with random premium.Then,the corresponding ruin probability is obtained.In chapter 3,based on the second chapter,we consider the recusive formula of expected discounted penalty function with random premium and dividends.Using the method of probability generation function,the corresponding ruin probability is obtained.In chapter 4,Chapter 4 takes Chapter 2 and Chapter 3 as starting points to further promote the model.We study the ruin probability of the discrete time semi-Markov model with two types of random dividends.At the same time,the influence of parameters on ruin probability is analyzed.