  This paper studies an optimal dividend payout problem with drawdown
constraint in a Brownian motion model, where the dividend payout rate must be
no less than a fixed proportion of its historical running maximum. It is a
stochastic control problem, where the admissible control depends on its past
values, thus is path-dependent. The related Hamilton-Jacobi-Bellman equation
turns out to be a new type of two-dimensional variational inequality with
gradient constraint, which has only been studied by viscosity solution
technique in the literature. In this paper, we use delicate PDE methods to
obtain a strong solution. Different from the viscosity solution, based on our
solution, we succeed in deriving an optimal feedback payout strategy, which is
expressed in terms of two free boundaries and the running maximum surplus
process. Furthermore, we have obtained many properties of the value function
and the free boundaries such as the boundedness, continuity etc. Numerical
examples are presented as well to verify our theoretical results and give some
new but not proved financial insights.
