  This paper studies a finite horizon utility maximization problem on excessive
consumption under a drawdown constraint up to the bankruptcy time. Our control
problem is an extension of the one considered in Bahman et al. (2019) to the
model with a finite horizon and also an extension of the one considered in Jeon
and Oh (2022) to the model with zero interest rate. Contrary to Bahman et al.
(2019), we encounter a parabolic nonlinear HJB variational inequality with a
gradient constraint, in which some time-dependent free boundaries complicate
the analysis significantly. Meanwhile, our methodology is built on technical
PDE arguments, which differs substantially from the martingale approach in Jeon
and Oh (2022). Using dual transform and considering some auxiliary parabolic
variational inequalities with both gradient and function constraints, we
establish the existence and uniqueness of the classical solution to the HJB
variational inequality and characterize all associated free boundaries in
analytical form. Consequently, the piecewise optimal feedback controls and some
time-dependent thresholds of the wealth variable for different control
expressions can be obtained.
