  This paper is concerned with a long standing optimal dividend payout problem
subject to the so-called ratcheting constraint, that is, the dividend payout
rate shall be non-decreasing over time and is thus self-path-dependent. The
surplus process is modeled by a drifted Brownian motion process and the aim is
to find the optimal dividend ratcheting strategy to maximize the expectation of
the total discounted dividend payouts until the ruin time. Due to the
self-path-dependent control constraint, the standard control theory cannot be
directly applied to tackle the problem. The related Hamilton-Jacobi-Bellman
(HJB) equation is a new type of variational inequality. In the literature, it
is only shown to have a viscosity solution, which is not strong enough to
guarantee the existence of an optimal dividend ratcheting strategy. This paper
proposes a novel partial differential equation method to study the HJB
equation. We not only prove the the existence and uniqueness of the solution in
some stronger functional space, but also prove the strict monotonicity,
boundedness, and $C^\infty$-smoothness of the dividend ratcheting free
boundary. Based on these results, we eventually derive an optimal dividend
ratcheting strategy, and thus solve the open problem completely. Economically
speaking, we find that if the surplus volatility is above an explicit
threshold, then one should pay dividends at the maximum rate, regardless the
surplus level. Otherwise, by contrast, the optimal dividend ratcheting strategy
relays on the surplus level and one should only ratchet up the dividend payout
rate when the surplus level touches the dividend ratcheting free boundary.
Moreover, our numerical results suggest that one should invest into those
companies with stable dividend payout strategies since their income rates
should be higher and volatility rates smaller.
