Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE

We consider the problem of maximizing the expected utility of discounted dividend payments of an insurance company. The risk process, describing the insurance business of the company, is modeled as Brownian motion with drift. We mainly consider power utility and special emphasis is given to the limiting behavior when the coefficient of risk aversion tends to zero. We then find convergence of the corresponding dividend strategies to the classical case of maximizing the expected dividend payments.