Stochastic optimal control of occupational pension funds

In this dissertation, we consider occupational pension funds that are set up by a company for its employees. A special feature of such pension funds is that the collectives of insured individuals are typically smaller, and benefit payments depend on the seniority and salary of the insured. Therefore, fluctuations over time in the composition of the collective of insured persons in terms of age, length of service, and salary cannot be ignored when calculating the actuarial liabilities and the company's financial contributions to the fund. We describe the stochastic dynamics of the composition of the collective of insured individuals by a discrete-time Markov chain model. This allows us to derive the dynamics of the expected aggregated actuarial liabilities for all insured individuals of the collective as a functional of the Markov chain. They are then approximated by mean-reverting processes, which facilitate the formulation of stochastic optimal control problems that arise in the cost-optimal management of occupational pension funds. Here, the objective is to invest the fund's capital in a financial market such that the expected net present value of all future contributions to the fund is minimized. The resulting optimal funding problem is studied under additional constraints that capture the obligations imposed by a regulator. They are intended to ensure that the fund capital is always sufficient to finance current benefit payments and future actuarial obligations. This problem is analyzed firstly under additional dynamic risk constraints and secondly under tracking constraints, which are included in the performance criterion with the help of penalties. While the risk constraints aim to keep the fund assets above the actuarial liabilities, the tracking constraints aim to keep these assets as close as possible to the liabilities. The resulting discrete-time stochastic optimal control problems are treated as Markov decision processes (MDPs). The corresponding Bellman equation for the value function is solved numerically using a backward recursion algorithm. We present the results of extensive numerical experiments showing the properties of the value function and the optimal decision rules, as well as their dependence on the various risk constraints.