  This paper studies the finite horizon portfolio management by optimally
tracking a ratcheting capital benchmark process. It is assumed that the fund
manager can dynamically inject capital into the portfolio account such that the
total capital dominates a non-decreasing benchmark floor process at each
intermediate time. The tracking problem is formulated to minimize the cost of
accumulated capital injection. We first transform the original problem with
floor constraints into an unconstrained control problem, however, under a
running maximum cost. By identifying a controlled state process with
reflection, the problem is further shown to be equivalent to an auxiliary
problem, which leads to a nonlinear Hamilton-Jacobi-Bellman (HJB) equation with
a Neumann boundary condition. By employing the dual transform, the
probabilistic representation and some stochastic flow analysis, the existence
of the unique classical solution to the HJB equation is established. The
verification theorem is carefully proved, which gives the complete
characterization of the feedback optimal portfolio. The application to market
index tracking is also discussed when the index process is modeled by a
geometric Brownian motion.

Portfolio Management;Optimization and Control;Probability