This paper studies a nonstandard stochastic control problem motivated by the
optimal consumption in an incomplete market with wealth tracking of a
non-decreasing benchmark process. In particular, the monotone benchmark is
modelled by the running maximum of a drifted Brownian motion. We consider a
relaxed tracking formulation using capital injection such that the wealth
compensated by the injected capital dominates the benchmark process at all
times. The stochastic control problem is to maximize the expected utility on
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
such that the singular control of capital injection and the floor constraint
can be hidden. To tackle the HJB equation with two Neumann boundary conditions,
we establish the existence of a unique classical solution to the dual PDE in a
separation form using some novel probabilistic representations involving the
dual reflected processes and the local time. The proof of the verification
theorem on the optimal feedback control can be carried out by some technical
stochastic flow analysis of the dual reflected processes and estimations of the
optimal control.Comment: Keywords: Non-decreasing benchmark, capital injection, optimal
consumption, Neumann boundary conditions, probabilistic representation,
reflected diffusion processe