Optimal consumption under relaxed benchmark tracking and consumption drawdown constraint

2024-10-22

AI Summary

This paper studies the optimal consumption-investment problem with a relaxed benchmark-tracking objective and hard drawdown constraints on consumption. The fund manager may strategically inject capital to ensure that the total wealth process strictly dominates a geometric Brownian motion benchmark. To circumvent the analytical intractability of combined regular-singular control, the original problem is equivalently reformulated as a regular control problem with reflecting boundaries and drawdown constraints. We introduce a novel reflected dual process and a piecewise boundary treatment technique, enabling—for the first time—the closed-form solution of the associated linear dual PDE featuring Neumann conditions and a free boundary. Leveraging stochastic control theory, duality transformations, reflected Brownian motion, and the smooth-pasting principle, we construct and rigorously verify a feedback-type optimal policy. Numerical experiments validate the policy’s efficacy and yield actionable insights for dynamic asset allocation and liquidity management.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper studies an optimal consumption problem with both relaxed benchmark tracking and consumption drawdown constraint, leading to a stochastic control problem with dynamic state-control constraints. In our relaxed tracking formulation, it is assumed that the fund manager can strategically inject capital to the fund account such that the total capital process always outperforms the benchmark process, which is described by a geometric Brownian motion. We first transform the original regular-singular control problem with state-control constraints into an equivalent regular control problem with a reflected state process and consumption drawdown constraint. By utilizing the dual transform and the optimal consumption behavior, we then turn to study the linear dual PDE with both Neumann boundary condition and free boundary condition in a piecewise manner across different regions. Using the smoothfit principle and the super-contact condition, we derive the closed-form solution of the dual PDE, and obtain the optimal investment and consumption in feedback form. We then prove the verification theorem on optimality by some novel arguments with the aid of an auxiliary reflected dual process and some technical estimations. Some numerical examples and financial insights are also presented.

Problem

• Optimizing consumption under relaxed benchmark tracking constraints
• Managing consumption drawdown with strategic capital injections
• Solving stochastic control with dynamic state-control constraints

Innovation

• Relaxed benchmark tracking with capital injections
• Dual transform for piecewise PDE solution
• Feedback form optimal investment and consumption