Optimal dividend payout under stochastic discounting

Adopting a probabilistic approach we determine theoptimal dividend payout policy of a firm whose sur-plus process follows a controlled arithmetic Brown-ian motion and whose cash-flows are discounted at astochastic dynamic rate. Dividends can be paid to share-holdersatunrestrictedratessothattheproblemiscastasoneofsingularstochasticcontrol.Thestochasticinterestrate is modeled by a Cox–Ingersoll–Ross (CIR) processand the firm’s objective is to maximize the total expectedflow of discounted dividends until a possible insolvencytime. We find an optimal dividend payout policy whichis such that the surplus process is kept below an endoge-nously determined stochastic threshold expressed as adecreasing continuous function↦()of the currentinterest rate value. We also prove that the value func-tion of the singular control problem solves a variationalinequality associated to a second-order, non-degenerateelliptic operator, with a gradient constraint.