Optimality studies for risk models with premium control

Abstract

This thesis studies two optimality problems for some risk models with premium control. In both problems, it is assumed that the basic insurance risk process follows a diffusion approximation to a non-homogeneous compound Poisson process, and that the insurance safety loading and the time-varying claim arrival rate are connected through a monotone decreasing function.

In the first problem, we investigate the optimal investment and reinsurance problem with premium control. In our set-up, the insurer can purchase proportional reinsurance to reduce potential risk, and is allowed to invest in a financial market which consists of one risky asset and one risk-free asset. It is assumed that the insurance and reinsurance safety loadings have a linear relationship. Applying the theory of stochastic control, we are able to derive the optimal strategy that maximizes the expected exponential utility of terminal wealth. We also provide a few numerical examples to examine the impact of the model parameters on the optimal strategy.

In the second problem, we consider the optimal dividend problem with premium control. In our problem formulation, both bounded and unbounded dividend rates are considered. Our aim is to find the optimal strategy that maximizes the expected discounted dividends until ruin. It is shown that the optimal strategy is a barrier strategy for unbounded dividend rates, and that the optimal strategy is a threshold strategy for bounded dividend rates. A few numerical examples are also given to show how the optimal strategies are affected by the model parameters.