Lundgren M. Mathematical investigation of insurance market with financial investments

The thesis is devoted to the investigation of risk processes which contain investments into risky and non-risky assets and variable premium intensity function. In the first chapter we make a short review of literature connected with the topics studied in the thesis. In the second chapter basic definitions and statements which are used in subsequent chapters are collected. The third chapter is dedicated to the risk model with investment of all capital into one type of risky assets and variable premium intensity function. We give sufficient conditions for existence of a strong solution of the risk process, and also sufficient conditions for the exponential process to be supermartingale. We find an exponential ruin probability estimate in the case of quadratic premium intensity function. For the model with stopping investments at the moment when price of risky asset falls below a fixed level ruin probability estimate was found in the case of locally Lipschitz continuous premium intensity function. The forth chapter deals with the risk model with investments into one type of risky assets and non-risky assets, and also variable premium intensity function. Sufficient conditions for an existence of a strong solution of the risk process are given. Exponential process is proved to be supermartingale under certain conditions, and exponential ruin probability estimate is found for an linearly growing intensity function. The obtained estimate improves estimates known for models without investments, and for models with investments but with constant premium intensity. It is proved that the problem of minimizing the ruin probability with the choice of a suitable investment strategy has a solution. The optimal strategy is calculated using Hamilton-Jacobi-Bellman equation. In the fifth chapter the risk model with variable premium intensity function and investments into finite number of risky assets and one non-risky asset is considered. Using supermartingale technique, an exponential ruin probability estimate in the case of Lipschitz continuous premium intensity function and constant investments into risky assets is found.