Optimal Ratcheting of Dividends in a Brownian Risk Model

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