  This paper studies stochastic control problems motivated by optimal
consumption with wealth benchmark tracking. The benchmark process is modeled by
a combination of a geometric Brownian motion and a running maximum process,
indicating its increasing trend in the long run. We consider a relaxed tracking
formulation such that the wealth compensated by the injected capital always
dominates the benchmark process. The stochastic control problem is to maximize
the expected utility of consumption deducted by the cost of the capital
injection under the dynamic floor constraint. By introducing two auxiliary
state processes with reflections, an equivalent auxiliary control problem is
formulated and studied, which leads to the HJB equation with two Neumann
boundary conditions. We establish the existence of a unique classical solution
to the dual PDE using some novel probabilistic representations involving the
local time of some dual processes together with a tailor-made
decomposition-homogenization technique. The proof of the verification theorem
on the optimal feedback control can be carried out by some stochastic flow
analysis and technical estimations of the optimal control.
