We consider an insurance company modelling its surplus process by a
Brownian motion with drift.

 In the first part of the talk, our target is to maximise the expected
exponential utility of discounted dividend payments, given that the
dividend rates are bounded by some constant.
Numerical and theoretical considerations lead us to the conclusion that
the optimal strategy must be of a non-trivial barrier
type. Our approach estimates the distance between the performance
function corresponding to a non-optimal (but easy-to-handle) strategy to
the value function.

In the second part, we look at several optimisation settings for an
insurance company under the constraint that the terminal surplus at a
deterministic and finite time T follows a normal distribution with given
mean and variance. We show when one can find explicit expressions for
the optimal strategies and the corresponding value functions. In
addition, the cases when the optimal strategy is not identified, will be
discussed and illustrated by examples.)