Optimal stochastic control of dividends and capital injections

In this thesis, we consider optimisation problems of an insurance company whose risk reserve process follows the settings of the classical risk model. The insurer has the possibility to control its surplus process by paying dividends to the shareholders. Furthermore, the shareholders are allowed to make capital injections such that the surplus process stays nonnegative. A control strategy describes the decision on the times and the amount of the dividend payments and the capital injections. To measure the risk associated with a control strategy, we consider the value of the expected discounted dividends minus the penalised expected discounted capital injections. Because of the discounting, it can only be optimal to make capital injections at times when the reserve would become negative due to a claim occurrence. Our goal is to determine the value function, which is defined as the maximal value over all proper strategies, and to find an optimal strategy which leads to this maximal value. First, we solve the optimisation problem for the classical risk model when the capital injections are penalised by some proportional factor greater than one. We show that an optimal strategy exists and is of barrier type, i.e., all the surplus exceeding some barrier level is paid as dividend. In the second part, we extend this model by adding fixed costs incurring any time at which capital injections are made. The optimal strategy here is not of barrier type any more, but of band type. That is a strategy where the state space of the surplus process is partitioned into three types of sets where either dividends at the premium rate, or lump sum dividends, or no dividends at all are paid. In the third part, we allow the dynamics of the surplus process to depend on environmental conditions which are modelled by a Markov process. I.e., the frequency of the claim arrivals and the distribution of the claim amounts vary over time depending on the state of the environment process. We again maximise the difference between the expected discounted dividends and the (proportional) penalised capital injections and show that the optimal strategy for any fixed initial environment state is of barrier type with a barrier depending on the initial state