  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 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 rate.
We also give a number of numerical illustrations in which the optimality of the
two-curve strategy can be established for instances with smaller values of the
maximum dividend rate, and the concrete form of the curves can be determined.
One observes that the resulting drawdown strategies nicely interpolate between
the solution for the classical unconstrained dividend problem and the one for a
ratcheting constraint as recently studied in Albrecher et al. (2022). When the
maximum allowed dividend rate tends to infinity, we show a surprisingly simple
and somewhat intriguing limit result in terms of the parameter $a$ for the
surplus level on from which, for sufficiently large current dividend rate, a
take-the-money-and-run strategy is optimal in the presence of the drawdown
constraint.
