On The Expected Discounted Penalty At Ruin

At the end of last century, the conception of the expected discounted penalty function at ruin was first introduced by Hans U. Gerber and Elias S. W. Shiu who are contemporary international leading experts at ruin theory. A number of particular cases of the expected discounted penalty function at ruin led to important quantities of interest in risk theory. These included the probability of ultimate ruin, the Laplace transform of the time of ruin, the joint and marginal distributions and moments of the surplus immediately before ruin and the deficit at ruin ,etc. The expected discounted penalty function at ruin being a powerful analytical tool made it possible to analyze the time of ruin, the surplus immediately before ruin, the deficit at ruin, and related quantities in a unified manner.By virtue of studying the properties and the application of the expected discounted penalty function at ruin, this dissertation is devoted to achieving three aspects of work as follows:1. In the classical risk model, Hans U. Gerber and Elias S. W. Shiu discussed the properties of the expected discounted penalty function at ruin in the case of constant force of interest, and from this obtained a lot of new results about the model. On this basis, we describe the interest randomness by standand Wiener process and Poisson process and obtain the renewal equation and the asymptotic formula for the expected discounted penalty function at ruin in this situation by using differential argument. By martingale approach, we get Lundberg's fundamental equation, and from this derive the probability of ultimate ruin and the probability that the surplus reaches the given level. Some results in classical risk theory are also discussed by virtue of this renewal equation, In the end, we obtain some results in the special case where the claim size obeys the exponential distribution.2. The compound Poisson risk model is considered in the presence of a threshold dividend strategy. In this model, no dividends are paid if the insurer's surplus is below certain threshold level and dividends are paid at a constant rate less than the premium rate when the surplus is above this threshold level. As to this model, we obtain the integro-differential equation and the analytical expression for the expected discounted penalty function at ruin in the case of constant force of interest. These results are utilized to derive the probability of ultimate ruin, theprobability of the first surplus drop below the initial level and the Laplace transform of the time of ruin. If the threshold level equals infinity, our risk model reduces to the classical model without constraints;If the threshold level equals the premium rate, our risk model coincides with the compound Poisson risk model under the constant barrier strategy, studied by X.Sheldon Lin, Gordon E. Willmot and Steve Drekic.3. We present a risk model with Poisson and Erlang (n) processes. In this model, the insurer have two dependent classes of insurance business for each of which the claim number process relate to Poisson process and the same Erlang (n) process. About this correlated aggregate claims risk model, the integro-differential equation and the Laplace transform for the expected discounted penalty function at ruin are obtained in the case of constant force of interest. Naturally, our results are generalization of those obtained by Kam C. Yuen, etc. and Shuanming Li etc. respectively in a risk model with Poisson and Erlang (2) processes and in the Erlang (n) risk model.