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The Study of the Risk Models in Markovian Environment

Author: WangChunWei
Tutor: YinChuanCun
School: Qufu Normal University
Course: Probability Theory and Mathematical Statistics
Keywords: Time of ruin Deficit at ruin Surplus prior to ruin Cox process Ruin probability Diffusion Martingale Stopping time
CLC: O211.67
Type: Master's thesis
Year: 2006
Downloads: 58
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Abstract


This dissertation is devoted to dealing with the ruin theory in Marko-vian environment. In most actuarial literatures related to risk theory, the assumption of independence between classes of business in an insurance book of business is made. In practice, however, there are situations in which this assumption is not satisfied. Papers that treat a relation of dependence between classes of business include Ambagaspitiya (1998), Baurble and Muller (1998), Cossette and Marceau (2000), Wu and Yuen (2003), Yuen, Guo and Wu (2002), Albrecher and Boxma (2004), etc. Among them, Cossette and Mareeau (2000) considered the Poisson model with common shock (PCS model) and the negative Binomial model with common component (NBCC model), Wu 和 Yuen (2003) also considered a discrete-time risk model risk with interaction between classes of business (IR (?)(?)del), Yuen, Guo and Wu (2002) discussed a correlated aggregate claims model with Poisson and Erlang risk processes. Albrecher and Boxma (2004) considered a generalization of the classical ruin model to a dependent setting, where the distribution of the time between twe claim occurrences depends on the previous claim size, and they derived the exact analytical expressions for the Laplace transform of the ruin function. In addition, because of the uncertainty of the payment or the income and the income of interest, so it leads to the risk model perturbed by diffusion and the risk model under interest and the Cox risk model. About the risk model under a constant interest, we can see Sundt and Teugels (1995), Sundt 和 Teugels (1997), Yang and Zhang (2001), etc. Sundt and Teugels (1995) discussed infinite time ruin probabilities in continuous time in a compound Poisson process with a constant premium rate and a constant interest rate, and discussed equations for the ruin probability as well as approximations and upper and lowerbounds. Sundt and Teugels (1997;discussed the risk model after they defined in 1995, and got more explicit information on the adjustment function. Yang and Zhang (2001) also considered the problem of the severity of ruin for the risk model defined by Sundt and Teugels (1995) and got the equations satisfied by the distributions of surplus immediately after ruin. There are many papers on the risk model perturbed by diffusion and the Cox risk model, wo refer to Dufresne and Gerber (1991), Gerber and Landry (1998), Tsai and .Willmot (2002), Tsai (2003), Jasiulewicz (2001), Wang (2001), Wang and Wu (2000) and so on. Gerber and Shiu (1998) introduced the discounted penalty function at ruin for the classical risk model. Cai and Dickson (2001) considered the discounted penalty function at ruin of a surplus process with interest. Albrecher and Boxma (2005) considered the discounted penalty function in h Markov-dependent risk model. Motivated by these literatures, in this dissertation, We’ll consider the discounted penalty function at ruin for the risk mode] in Markovian environment. Furthermore, we also consider the risk model in Markovian environment with interest,.In Chapter 1, we mainly consider a Markov-modulated risk model in which the claim inter-arrivals and claim sizes are influenced by an external Markovian environment process. We analyze the discounted penalty function in the risk model by means of Laplace-Stieltjes transforms, we derive its explicit expression when the initial capital is zero, its asymptotic behavior and moments of the characteristics of the risk process.In Chapter 2, we consider a Markov-modulated risk model in which the claim inter-arrivals and claim sizes are influenced by an external Markovian environment process with interest. Furthermore, we derive an integral equation based on the analysis of the discounted penalty function for the model. We obtain two equations by means for the Laplace transforms of the discounted penalty function.In Chapter 3, we generalize the classical risk model that is perturbed by diffusion, that is, the arrival of the claims is a Cox process, and the arrival oi the premium income process is a Poisson process besides the constant premium rate c. By the method of martingale, we get the inequality for the ultimate ruin probability, and give an explicit expression when there is no perturbation as well as the arrival of the policies and the occurrence of the claims have the same Poisson process.

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CLC: > Mathematical sciences and chemical > Mathematics > Probability Theory and Mathematical Statistics > Theory of probability ( probability theory, probability theory ) > Random process > Expectations and Forecast
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