Some stochastic control problems in insurance

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Stochastic control has been used in insurance for some time for peculiar problems (see Martin-Lof [7], or Brockett[2]). In this paper we shall consider continuous time problems which lead to Hamilton-Jacobi-Bellman equations. These are problems of optimal choice of new business, and of optimal proportional reinsurance. Our objective function will be infinite time ruin probability (which is chosen for simplicity and for the purpose of illustration). Other objective functions (based on utility functions or on expected discounted dividend) are possible and can be treated with essentially the same methods. See e.g. Asmussen and Taksar [1] or Hoejgaard and Taksar [6].