Optimal dividends under a stochastic interest rate

We consider an insurance entity endowed with an initial capital and an income, modelled as a Brownian motion with drift. The discounting factor is modelled as a stochastic process: at first as a geometric Brownian motion, then as an exponential function of an integrated Ornstein-Uhlenbeck process. It is assumed that the insurance company seeks to maximize the cumulated value of expected discounted dividends up to the ruin time. We find an explicit expression for the value function and for the optimal strategy in the first but not in the second case, where one has to switch to the viscosity ansatz.