  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 finite continuum of admissible dividend
rates we establish the $\varepsilon$-optimality of curve strategies. This work
is a counterpart of Albrecher et al. (2020), where the ratcheting problem was
studied for a compound Poisson surplus process with drift. In the present
Brownian setup, calculus of variation techniques allow to obtain a much more
explicit analysis and description of the optimal dividend strategies. We also
give some numerical illustrations of the optimality results.
