Abstract

We study a nonstandard stochastic control problem motivated by the optimal consumption 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, and a homogenization technique of Neumann boundary conditions. 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.

Biography

薄立军，教授。本科毕业于西安电子科技大学、硕士和博士毕业于南开大学概率论与数理统计专业，研究方向为随机分析、随机控制与金融数学。先后主持国家自然科学基金青年、面上、教育部新世纪优秀人才支持计划、中科院前沿科学重点以及陕西国家应用数学中心交叉团队培育项目等。目前已在Ann. Appl. Probab.、Math. Finan.、Stoch. Process. Appl.、SIAM J. Contr. Optim.、SIAM J. Finan. Math.、Math. Opers. Res.、J. Banking Finan.等学术期刊发表学术论文70余篇，出版本科和研究生教材四部。