Optimal investment for investors with state dependent income, and for insurers

An optimal control problem is considered where a risky asset is used for investment, and this investment is financed by initial wealth as well as by a state dependent income. The objective function is accumulated discounted expected utility of wealth, where the utility function is nondecreasing and bounded. This problem is investigated for constant as well as for stochastic discount rate, where the stochastic model is a time homogeneous finite state Markov process. We prove that the Bellman equation to this optimization problem has a classical solution and give a verification argument. Based on this we deal with the problem of optimal investment for an insurer with an insurance business modelled by a compound Poisson or a compound Cox process, under the presence of constant as well as (finite state space Markov) stochastic interest rate.