Ruin Probability Minimization and Dividend Distribution Optimization in Diffusion Models

We consider a model of an insurance companies with different modes of risk and financial control. Different types of reinsurance correspond to the risk reduction techniques of the insurance, while financial control corresponds to a more familiar portfolio rebalancing. There are different objective which the company may pursue. One is the classical minimization of the ruin probabilities. Another one is the dividend pay-out maximization. The later merges with the classical finance issue of utility optimization by a small investor, pioneered by Merton. Diffusion approximation enables one to get a closed form solution to many problems and see the structure of the optimal policy. Mathematically, the problem becomes a mixed singular/regular control of a diffusion process, whose analytical portion corresponds to a solution of nonlinear ordinary or partial differential equations