Optimal Ratcheting of Dividends in a Brownian Risk Model

Abstract

We study the problem of optimal dividend payout from a surplus process governed by Brownian motion with drift under the additional constraint of ratcheting, i.e., the dividend rate can never decrease. We solve the resulting two-dimensional optimal control problem, identifying the value function to be the unique viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation. For finitely many admissible dividend rates we prove that threshold strategies are optimal, and for any closed interval of admissible dividend rates we establish the $\varepsilon$-optimality of curve strategies. This work is a counterpart of [H. Albrecher, P. Azcue, and N. Muler, SIAM J. Control Optim., 58 (2020) pp. 1822--1845], where the ratcheting problem was studied for a compound Poisson surplus process with drift. In the present Brownian setup, calculus of variation techniques allow us to obtain a much more explicit analysis and description of the optimal dividend strategies. We also give some numerical illustrations of the optimality results.