Stable dividends under linear-quadratic optimisation

Abstract

The optimisation criterion for dividends from a risky business is most often formalised in terms of the expected present value of future dividends. That criterion disregards a potential, explicit demand for the stability of dividends. In particular, within actuarial risk theory, the maximisation of future dividends has been studied as the so-called de Finetti problem. However, there the optimal strategies typically become so-called barrier strategies. These are far from stable, and suboptimal affine dividend strategies have recently received attention. In contrast, in the class of linear-quadratic problems, the demand for stability is explicitly stressed. These have often been studied in diffusion models different from the actuarial risk models. We bridge the gap between these thinking patterns by deriving optimal affine dividend strategies under a linear-quadratic criterion for an additive process. We characterise the value function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal controls to the classical objective of maximising the expected present value of future dividends. Thereby we provide a framework within which stability of dividends from a risky business, e.g. in classical risk theory, is explicitly demanded and obtained.