Equilibrium Strategies for Singular Dividend Control Problems under the Mean-Variance Criterion

Abstract. We revisit the optimal dividend problem of de Finetti by adding a variance term to the usual criterion of maximizing the expected discounted dividends paid until ruin, in a singular control framework. Investors do not like variability in their dividend distribution, and the mean-variance (MV) criterion balances the desire for large expected dividend payments with small variability in those payments. The resulting MV singular dividend control problem is time-inconsistent, and we follow a game-theoretic approach to find a time-consistent equilibrium strategy. Our main contribution is a new verification theorem for the novel dividend problem, in which the MV criterion is applied to an integral of the control until ruin, a random time that is endogenous to the problem. We demonstrate the use of the verification theorem in two cases for which we obtain the equilibrium dividend strategy (semi)explicitly, and we provide a numerical example to illustrate our results.