Optimal Tracking Portfolio with a Ratcheting Capital Benchmark

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