Optimal dividends under a drawdown constraint and a curious square-root rule

Abstract

In this paper, we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint, i.e., the dividend rate can never decrease below a given fraction a of its historical maximum. We solve the resulting two-dimensional optimal control problem and identify the value function as the unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We then derive sufficient conditions under which a two-curve strategy is optimal, and we show how to determine its concrete form using calculus of variations. We establish a smooth-pasting principle and show how it can be used to prove the optimality of two-curve strategies for sufficiently large initial and maximum dividend rates.