SIAM Journal on Control and Optimization, Volume 59, Issue 3, Page 2346-2380, January 2021.
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 capital into the portfolio account such that the total capital dominates a nondecreasing 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, but 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 a unique classical solution to the HJB equation is established. The verification theorem is carefully proved, which gives a 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.

中文翻译：

---

具有棘轮资本基准的最佳跟踪投资组合

SIAM Journal on Control and Optimization，第 59 卷，第 3 期，第 2346-2380 页，2021 年 1 月。
本文通过优化跟踪棘轮资本基准过程来研究有限期限投资组合管理。假设基金经理可以动态地将资本注入投资组合账户，使得总资本在每个中间时间主导一个非递减的基准下限过程。制定跟踪问题以最小化累积资本注入的成本。我们首先将具有楼层约束的原始问题转化为无约束控制问题，但在运行最大成本下。通过识别具有反射的受控状态过程，该问题进一步显示为等价于辅助问题，这导致非线性 Hamilton--Jacobi--Bellman (HJB) 方程具有 Neumann 边界条件。通过使用对偶变换，概率表示，和一些随机流动分析，建立了 HJB 方程的唯一经典解的存在性。验证定理经过仔细证明，给出了反馈最优投资组合的完整表征。当指数过程由几何布朗运动建模时，还讨论了市场指数跟踪的应用。