Application Of The Markov Process In Discrete Risk Model

This article, based on the compound binomial risk model, clued by the Markov process, is applied the Markov process to all aspects so as to establish a new model. We modify the model by adding a variety of factors such as Markov process, investment, dividend, Markovian environment and so on. In thesis, we mainly use Markov process, probability and statistics, stochastic processes, combinatorial mathematics, matrix, economic theory, the renewal theory, the stochastic control theory to obtain the Gerber-Shiu function or the expectation of discounted dividends until ruin.The application of the Markov process in discrete risk model is studied in this paper. The main problems studied in this paper are as follows.1. Based on a compound binomial model, we embed two factors, i.e. the periodic dividend and Markovian environment. We assume the premium rate, probability of the claim occurs, the claim amount and the dividend barrier are all influenced by the Markovian environment. At the same time, insurers consider giving dividends to shareholders at a periodic time. Some properties about this model are derived. A system of integral equations for the expecta-tion and the rth moment of discounted dividends until ruin time are obtained respectively. Moreover, we solved the equation system and obtained the explicit expression.This chapter is published in the Acta Mathematica Sinica, English series,2015(2).2. Based on a compound binomial model, we try to use an irreducible and homogeneous discrete-time Markov chain to modulate the inter-observation times and embed a dividend strategy. Therefore, a Markov observation model with dividend is defined. In a recent survey, the company is only checked on a periodic basis. We try to use an irreducible and homogeneous discrete-time Markov chain to modulate the inter-observation times and call it Markov observation model. In the Markov observation model with dividend, a system of liner equations for the expected discounted value of dividends until ruin time is derived. Moreover, an explicit expression is obtained and proved. Finally, some interesting properties are illustrated by numerical analysis and by comparing with the complete compound binomial model with dividend.3.We use Markov arrival process to depict claims arrival process and define the Markov arrival risk model. Discrete markov arrival process is widely used in the queuing theory, and is a very general discrete markov process. And on the basis of the Markov arrival risk model, we think with the barrier dividend. The Gerber -Shiu function and total expected discounted dividend before ruin time is considered. We get the equations for the Gerber-Shiu function and discounted dividend. Moreover, the calculation method and the exact expression are obtained,respectively. Finally, we give numerical solution to make explanations.4. Markov arrival model in the problem 3 is modified to batch Markov arrival model, that is the Markov arrival process is modified to the batch Markov arrival process. At the same time, the premium rate and the payouts process was controlled by the phase process. Thus, we get the batch Markov arrival risk model. Finally, the iterative formula for ruin probability and numerical algorithm are studied.5. Based on the model in the previous problem, we add the the barrier dividend and obtain the batch Markovian arrival model with dividend. Re-cursion formulas of the Gerber-Shiu function and the first discounted dividend value are provided and the expressions of the total discounted dividend value are obtained and proved. At the last part, some numerical illustrations was presented.6. We mainly study the compound binomial model with investment costs as well as considering proportion investment costs. We want to make the minimum variance, at the same time, expecting to achieve the given value in the final time. Finally, the expressions of the optimal investment strategy and the value function are derived respectively.7. We discussed a compound binomial risk model with the market invest-ment and uncertain time exit, all modulated by a Markov process. That is abased on the compound binomial investment model, we embed the Markovian environment and uncertain time exit. We study the mean-variance problem. The optimal investment strategy and value function of expression of the auxil-iary model Ai(λ,ω)and the original problem Pi(ω) is obtained,respectively.In this article, we mainly make innovative application of Markov process into three big aspects:The first aspect is mainly for periodic problem. It includes periodic div-idends and periodic observation. In the second chapter, we firstly embed a Markov process into a model with periodic dividend. In the third ,chapter, we firstly use a Markov process to control the random observation interval, and establish the Markov observation risk model with barrier dividend.The second aspect is mainly about application for batch Markovian arrival process, which is a special and importane kind of markov process. In the fourth chapter, the fifth chapter and chapter 6, we use batch Markovian arrival process instead of binomial process. We study it, from shallow to deep,from easy to difficult, from special to gradual.Third aspect is mainly for the optimal mean-variance control problem. In chapter 7, the Markov process is introduced into the wealth process with trans-action cost. The optimal investment strategy and value function is studied. In chapter 8, the Markov process can be represented as the environment condition, and control the exiting time, the rate of return on risky assets, probability of the claim occurs, the claim amount. The optimal investment strategy and the value function of the variance problem are studied,respectively.