Stochastic control problems with state-reflections arising from relaxed benchmark tracking

TL;DR

This paper addresses stochastic control problems involving wealth benchmark tracking with state reflections, proposing a novel approach to solve the associated HJB equations and establish optimal control strategies.

Contribution

It introduces a new formulation with auxiliary reflected processes and develops a probabilistic method to solve the dual PDE for optimal control.

Findings

Existence of a unique classical solution to the dual PDE.

Development of a probabilistic representation involving local time.

Verification of the optimal feedback control through stochastic flow analysis.

Abstract

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.