Markovian Dual Risk Model With Random Observation

The study of the classical risk model and the dual risk model has received remarkable attention. The classical risk model is also called Cram′er-Lundberg risk model. Many generalized classical risk models have been studied such as adding the interest into models,stochastic premiums etc. The Markov-modulated risk model is a generalized risk model.A dual risk model is a risk model which the premiums are regarded as costs and the claims are viewed as profits. In the Markov-modulated dual risk model, the rate of the Poisson income arrival process and the distribution of the income sizes change as the state of an underlying Markov jump process {J(t) : t ≥ 0}. In the aboved risk model,we set an upper dividend barrier b and then decide whether to pay dividends to the shareholders by comparing the size of the surplus value and b. In reality, it is impossible to observe the surplus value all time continuously. So it may be more reasonable to consider the situation with randomized observation, The ideal of randomized observation means the risk process can be ”looked” only at random times(called observation times), which naturally lead to the study of discrete-time risk model. In this paper, we discuss some issues in a markov-modulated dual risk model with randomized observation. According to the research details, this paper is organized as follows:In chapter one, we introduce the history of the risk model. Then we introduce the classical risk model, the dual risk model and the Markov-modulated risk model.In second chapter, we first introduce the markov-modulated dual risk model with randomized observation that researched in this paper. And, given that the initial environment state is i(i = 1, 2, · · ·, m), we derive a system of integro-differential equations satisfied by Vi(u; b). For arbitrary i(i = 1, 2, · · ·, m), Vi(u; b) represents the expected present value of the dividend payments until ruin. Secondly, we give a matrix formal expression of the integro-differential equations satisfied by Vi(u; b) by applying the operator(d/du- vi). At last, we study the discounted density of increment gδ(y).