A Variational Approach for Mean-Variance-Optimal Deterministic Consumption and Investment

A significant number of life insurance contracts are based on deterministic investment strategies—this justifies to restrict the set of admissible controls to deterministic controls. Optimal deterministic controls can be identified by Hamilton-Jacobi-Bellman techniques, but for the corresponding partial differential equations only numerical solutions are available and so the general existence of optimal controls is unclear. We present a non-constructive existence result and derive necessary characterizations for optimal controls by using a Pontryagin maximum principle. Furthermore, based on the variational idea of the Pontryagin maximum principle, we derive a numerical optimization algorithm for the calculation of optimal controls. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.