This paper studies the finite horizon portfolio management by optimally tracking a ratcheting capital benchmark process. To formulate such an optimal tracking problem, we envision that the fund manager can dynamically inject capital into the portfolio account such that the total capital dominates the nondecreasing benchmark floor process at each intermediate time. The control problem is  to minimize the cost of the accumulative 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, we next transform the problem further into an equivalent auxiliary problem, which leads to a nonlinear Hamilton-Jacobi-Bellman (HJB)  with a Neumann boundary condition. By employing the dual transform, the probabilistic representation approach and some stochastic flow arguments , the existence of the unique classical solution to the dual HJB  is established. The verification theorem is carefully proved, which gives the complete characterization of the primal value function and the feedback optimal portfolio  .
This paper studies the finite horizon portfolio management by optimally tracking a ratcheting capital benchmark process. We envision that the fund manager can dynamically inject capital into the portfolio account such that the total capital dominates the non-decreasing benchmark floor process at each intermediate time. The control 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, we next transform the problem further into an equivalent 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 dual HJB equation is established. The verification theorem is carefully proved, which gives the complete characterization of the primal value function and the feedback optimal portfolio . The application to market index tracking is also discussed when the index process is modeled by a geometric Brownian motion .
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