Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate

We investigate a dividend maximization problem under stochastic interest rates with Ornstein-Uhlenbeck dynamics. This setup also takes negative rates into account. First a deterministic time is considered, where an explicit separating curve $\alpha(t)$ can be found to determine the optimal strategy at time $t$. In a second setting we introduce a strategy-independent stopping time. The properties and behavior of these optimal control problems in both settings are analyzed in an analytical HJB-driven approach as well as using backward stochastic differential equations.