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

TL;DR

This paper extends the classical dividend problem by incorporating a variance term into the criterion, leading to a time-inconsistent control problem that is addressed through a game-theoretic equilibrium approach, with explicit solutions and numerical illustrations.

Contribution

It introduces a new verification theorem for a mean-variance dividend control problem with an endogenous ruin time, providing explicit equilibrium strategies.

Findings

Derived semi-explicit equilibrium dividend strategies.

Established a new verification theorem for the mean-variance control problem.

Provided numerical examples illustrating the equilibrium strategies.

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.