  This paper studies a life-time consumption-investment problem under the
Black-Scholes framework, where the consumption rate is subject to a lower bound
constraint that linearly depends on her wealth. It is a stochastic control
problem with state-dependent control constraint to which the standard
stochastic control theory cannot be directly applied. We overcome this by
transforming it into an equivalent stochastic control problem in which the
control constraint is state-independent so that the standard theory can be
applied. We give an explicit optimal consumption-investment strategy when the
constraint is homogeneous. When the constraint is non-homogeneous, it is shown
that the value function is third-order continuously differentiable by
differential equation approach, and a feedback form optimal
consumption-investment strategy is provided. According to our findings, if one
is concerned with long-term more than short-term consumption, then she should
always consume as few as possible; otherwise, she should consume optimally when
her wealth is above a threshold, and consume as few as possible when her wealth
is below the threshold.
