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

Paper: arXiv 2302.08302

Authors: Lijun Bo, Yijie Huang, Xiang Yu

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.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents extremely rigorous mathematical derivations in stochastic control theory, Hamilton-Jacobi-Bellman (HJB) equations, and state-reflection problems. However, it is entirely theoretical with no empirical backtesting, financial datasets, or applied simulation studies included.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Consumption with<br/>Relaxed Benchmark Tracking] --> B[Inputs: Wealth Process, Capital Injection &<br/>Benchmark: GBM + Running Max]
    B --> C[Formulate Stochastic Control Problem<br/>with Dynamic Floor Constraint]
    C --> D[Transform to Auxiliary Control Problem<br/>with Two Reflected State Processes]
    D --> E[Derive HJB Equation with<br/>Two Neumann Boundary Conditions]
    E --> F[Solve Dual PDE via Local Time<br/>& Decomposition-Homogenization]
    F --> G[Outcomes: Unique Classical Solution &<br/>Verification of Optimal Feedback Control]