This paper studies stochastic control problems motivated by optimal consumption with wealth benchmark tracking. The benchmark process is modeled by a combination of a geometric Brownian motion and a running maximum process, indicating its increasing trend in the long run. We consider a relaxed tracking formulation such that the wealth compensated by the injected capital always dominates the benchmark process. The stochastic control problem is to maximize the expected utility of 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, which leads to the HJB equation with two Neumann boundary conditions. We establish the existence of a unique classical solution to the dual PDE using some novel probabilistic representations involving the local time of some dual processes together with a tailor-made decomposition-homogenization technique. The proof of the verification theorem on the optimal feedback control can be carried out by some stochastic flow analysis and technical estimations of the optimal control.
A Relaxed Quadratic-Program-based Framework for Trajectory Tracking of Unicycle Robots with Singularity Avoidance
Department of Mechanical；Industrial Engineering at the New Jersey Institute of Technology (NJIT), NJ, USA；School of Engineering, Infrastructure；Sustainability De Montfort University, Leicester, LE1 9BH, UK
Mean Field Game of Controls with State Reflections: Existence and Limit Theory
School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
