  In this paper, we study two 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 a given mean and a given variance.
In both cases, the surplus of the insurance company is assumed to follow a
Brownian motion with drift. First, we allow the insurance company to pay
dividends and seek to maximise the expected discounted dividend payments or to
minimise the ruin probability under the terminal distribution constraint. Here,
we find explicit expressions for the optimal strategies in both cases: in
discrete and continuous time settings. Second, we let the insurance company buy
a reinsurance contract for a pool of insured or a branch of business. To
achieve a certain level of sustainability (i.e. the collected premia should be
sufficient to buy reinsurance and to pay the occurring claims) the initial
capital is set to be zero. We only allow for piecewise constant reinsurance
strategies producing a normally distributed terminal surplus, whose mean and
variance lead to a given Value at Risk or Expected Shortfall at some confidence
level $\alpha$. We investigate the question which admissible reinsurance
strategy produces a smaller ruin probability, if the ruin-checks are due at
discrete deterministic points in time.

Mathematical Finance;Probability;Risk Management