Two dimensional optimization problems in Actuarial Mathematics

An insurance company, planning its business for the next couple of years, has first of all to evaluate its current state. In classical Actuarial Mathematics two approaches are mainly used. The first one, going back to F. Lundberg, tries to estimate the probability for an eventual ruin of the company. In order to do this, one has first to establish a probabilistic model for the time evolution of the wealth of the company. Dependent on the chosen model, one is able to give either explicit expressions, approximative values or upper estimates for the ruin probability. The second approach goes back to the Italian actuar B. de Finetti. He considered the ruin probability approach as being too conservative and suggested an alternative. His approach uses the maximal discounted dividend payments an insurance company is able to pay its shareholders in the future. Again, depending on the nature of the probabilistic model of the endowment process, more or less explicit results for the optimal dividend strategy on the one hand, and for the maximal possible payments on the other hand are possible. What we have described so far concerns the one-dimensional situation, i.e. we consider one insurance company and its wealth process. Nowadays it is very well understood. This is certainly not true for the higher dimensional case, i.e., if one tries to estimate ruin probabilities or optimal dividend payments for several companies. Only very recently there have been some publications in this direction. Our aim is to contribute to the literature in this field and to focus first of all on the two dimensional case. Let us finally describe very briefly one specific problem, namely the dividend optimization problem for two companies. In this model the companies are not only allowed to pay dividends to its shareholders, but they can collaborate. This means, e.g., that, if the situation becomes critical for company A, company B is allowed to help by transfer payments. The aim is to maximize the aggregated value of the dividend payments of both companies. The mathematical problem is now, to find out the optimal instances for these payments. Similar problems can be formulated for the ruin probability approach as well.

An insurance company which plans its business activities for the future should first analyze the current situation. In actuarial mathematics, two main approaches are used. The first goes back to F. Lundberg and tries to estimate the ruin probability of the company. Here it is necessary to develop a probabilistic model for the time development of the company's wealth. The second approach was introduced by the Italian actuary B. de Finetti, to whom the above approach seemed to be too conservative. His alternative: The company should be evaluated by how much dividends it can pay to its shareholders in the future. These considerations only concern the one-dimensional situation, i.e. one deals with one insurance company and its wealth process. In our project we investigated the case of two companies that are allowed to collaborate, which is not well studied in the literature. This means that if the situation for company A becomes critical, company B is allowed to help with transfer payments. The aim of both companies is to minimize the ruin probability. In our simplest model, it is optimal that the company with currently lower wealth is maximally supported by the other company. For more general models we also provide an optimal strategy. In some cases, the minimal ruin probability can be calculated explicitly. If transfer payments are made in order to maximize the total dividend payments, we can explicitly describe the optimal strategy in our model, i.e. the amount of transfer payments and the dividends at any point in time.