Optimal control models with different risk measures, infinite or finite time horizon play an important role in actuarial mathematics. By such problems, one considers a target functional that quantifies risks connected to an insurance portfolio augmented by additional components, such as dividend payments, investments, reinsurance etc. The main goal is to find a strategy, that minimizes/maximizes the underlying target functional, i.e. minimizes the risks or maximizes the benefits of the insurer. Thus, the choice of a risk measure is of crucial importance for solving control problems.
The next step, when formulating a control problem, is the modeling of the surplus process of the considered insurance entity. Here, one has the choice between deterministic and stochastic modeling. Most economic problems are of a stochastic nature, due to the uncertainty about future system development. On the one hand, stochastic models approximate the real processes much better than deterministic ones, on the other - deterministic modeling enjoys a much greater ease of computability. However, it should be noticed that even in a deterministic setting it is only rarely possible to find a closed expression for the optimal strategy.
The present project aims to consider three different risk measures: expected discounted consumption, capital injections and dividends under deterministic and stochastic surplus modeling.
The first model describes the situation when the surplus of an economic agent is of deterministic nature. As an example one may think of households living from tourism (every summer regular income, every winter ``employment gap''). The discounting factor is stochastic (depends on the global macroeconomic situation, which cannot be assumed to be deterministic). We target to maximize the expected discounted consumption of the agent up to a finite deterministic time horizon.
The second problem models the surplus of the considered insurance company by a Brownian motion with drift. In addition, the insurer is allowed to buy reinsurance and pay out dividends. It is also assumed, that in the case the surplus process becomes negative, the shareholders have to inject capital in order to facilitate a smooth business progress, which allows to consider the problem in the infinite time horizon. The model parameters, such as the drift, the volatility, the retention level of the reinsurance, the safety loadings of the insurer and reinsurer, evolve due to a Markov process. The main goal is to find a reinsurance strategy, depending on the initial capital and initial state of the Markov process, that would maximize the value of expected difference of discounted dividends and capital injections.
Finally, we want also to contribute in the direction of maximizing dividend payments. Here, we again model the insurer's surplus by a Brownian motion with drift and want to maximize the expected discounted dividend payments under a finite time horizon. If the surplus process remains non-negative until the final time, the remaining surplus becomes the final payment, i.e. we pay out whatever is left. On the other hand, too much consumption is penalized only through the fact that the insurer will be ruined earlier. In our model, the insurer will be additionally rewarded if he ¿stays alive¿ until the final time.
