  We consider a discrete-time version of the popular optimal dividend pay-out
problem in risk theory. The novel aspect of our approach is that we allow for a
risk averse insurer, i.e., instead of maximising the expected discounted
dividends until ruin we maximise the expected utility of discounted dividends
until ruin. This task has been proposed as an open problem in H. Gerber and E.
Shiu (2004). The model in a continuous-time Brownian motion setting with the
exponential utility function has been analysed in P. Grandits, F. Hubalek, W.
Schachermayer and M. Zigo (2007). Nevertheless, a complete solution has not
been provided. In this work, instead we solve the problem in discrete-time
setup for the exponential and the power utility functions and give the
structure of optimal history-dependent dividend policies. We make use of
certain ideas studied earlier in N. B\"auerle and U. Rieder (2013), where
Markov decision processes with general utility functions were treated. Our
analysis, however, include new aspects, since the reward functions in this case
are not bounded.
