A Study On The Absolute Ruin Problem In A Two Classes Of Risk Processes

The theoretical foundation of ruin theory, known as the classical compound-Poisson risk model in the literature, was introduced in1903by the Swedish actuary Filip Lundberg. The classical model was later extended to relax assumptions about the inter-claim time distribution, the distribution of claim sizes, etc. In most cases, the principal objective of the classical model and its extensions was to calculate the probability of ultimate ruin. Ruin theory received a substantial boost with the articles of Powers[12] in1995and Gerber and Shiu[11] in1998, which introduced the expected discounted penalty function, a generalization of the probability of ultimate ruin. This fundamental work was followed by a large number of papers in the ruin literature deriving related quantities in a variety of risk models.Consider a risk model with two classes of business is considered, in which claim number processes are modeled by two independent classes of insurance risk. We assume that the two claim number processes are independent Poisson and generalized Erlang(2) processes. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay debts from her premium income. The negative surplus may return to a positive level if debts are reasonable. However, when the negative surplus is below some critical level, the surplus is no longer able to be positive. Absolute ruin is said to occur at this moment.In this paper, we study absolute ruin questions by defining an expected dis-counted penalty function at absolute ruin or the Gerber-Shiu function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just be-fore absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the Gerber-Shiu func-tion. Then, we derive a system of renewal equations satisfied by the Gerber-Shiu function with initial surplus above zero via the Dickson-Hipp operator. Moreover, analytical solution of the systems of integro-differential equation are presented. Finally, we give explicit expressions for the Gerber-Shiu function at absolute ruin in a special case.