Some Problems Of Dual Model With Random Observation Time

The problems of dividend and ruin in risk model are always important subjects in the?eld of actuary. In this paper, we consider the problems of dividend and ruin under the dual risk model with random observation time. We consider dividend and ruin under the situation in which the gain amounts is the mixtures of exponential distribution and the dual model with dependence between gain intervals and gain amounts, respectively. The main dividend policy involved in this paper is the barrier strategy. The main actuarial quantities in this paper are expected discounted value of dividend and ruin probability. And the main tools used in this paper include the theory of ordinary di?erential equations, Markovian theory and matrix theory. According to the content of this paper, we divided it into the following three chapters:In Chapter 1, we mainly describe the history and development of the risk theory, various dividend policies and the research results under di?erent risk models. We also describe the model and give some comments on relevant quantities which will be researched in this paper.In Chapter 2, we discuss the problems of dividend and ruin in the dual model in which dividend and ruin can only be observed at these random observation times. First of all,we divide the range of initial surplus into(-∞, 0),(0, b),(b, ∞) three parts according to the impacts of random observations. Because of Markovian property of risk processes, we derive the integro-di?erential equations for the expected discounted dividend V(x; b) and ruin probability φ(x; b) with the help of the approach of conditioning on events in a small time interval on the three parts. Then the theory of ordinary di?erential equation can be used to derive the solutions of the integro-di?erential equations.In the last chapter, we discuss the problems of dividend and ruin under the dual model with dependence between gain intervals and gain amounts. In this model, dividend still can be only observed at these random observation times. The dividend policy is barrier strategy. Once the surplus level is below 0, the bankruptcy immediately happens. Using the same method which used in the second chapter, we derive the integro-di?erential equations of expected discounted dividends V(x; b) and the ruin probability φ(x; b) under this risk model. Then, we give the Laplace transform when the initial surplus between 0 and the level of barrier b on the basis of the equations.