Summary
This thesis consists, in excess to an introductory chapter, of five papers within the broad area of life insurance mathematics. The last of the papers can also be considered as within the area of non-life insurance mathematics. There is no unifying topic, but many recurrent subtopics.

In the first paper, we consider a market where the interest rate and mortality intensity are modelled as ane processes. The market consists of zero coupon bonds and longevity bonds and we do not assume that the price processes are martingales under the real measure. In this setup we find an optimal hedging strategy for a portfolio of general life insurance contracts by using the criterion of local risk-minimization.

The second paper studies expected policyholder behavior in a multistate Markov chain model with deterministic intensities. Valuation techniques in the cases where policyholder behavior is modelled to occur independently or dependently of insurance risk, respectively, are discussed and studied numerically. The impact (quantitatively and computationally) of dierent simplifying assumptions are investigated for representative insurance contracts.

In the third paper, we derive worst-case scenarios and reserves in a life insurance model in the case where the interest rate and the various transition intensities are mutually dependent. The calculations are based on deterministic optimal control theory. The results of a single insurance contract are extended to inhomogeneous portfolios of insurance contracts and various numerical
studies are presented. These studies qualify the standard formula of Solvency II.

In the fourth paper, we study the class of ane processes. We obtain transform results which can be used for valuation of life insurance contracts modelled within general, hierarchical, multistate Markov chains. The ane setup makes the calculations computationally tractable because we only need to solve systems of ordinary dierential equations and not partial dierential equations. The setup allows for mutual dependence between interest rate and transition intensities which makes it possible to e.g. model interest rate dependent surrender rates.

Finally, the fth paper obtains optimal surplus distribution strategies in a model with innite time horizon where assets and liabilities are modelled by correlated, geometric Brownian motions. The controls considered are, that we either increase liabilities or decrease assets. The increase of liabilities could be used in the modelling of non-for-prot mutual funds or pension funds. On the other hand, the decrease in assets could be used for modelling of for-prot companies. We
impose a simple solvency constraint and prove optimality of barrier strategies, where the barrier is defined in terms of the funding ratio. We also study barrier strategies within a model with a more advanced solvency constraint, where the allowance of controlling either liabilities or assets, when the funding ratio is between a lower and upper barrier, depends on which of the two barriers that have been crossed last. We also consider barrier strategies under the assumption that ruin must be prevented which is done by either decreasing the liabilities or by capital injections. All the results are illustrated numerically.