Optimal dividend payout with path-dependent drawdown constraint

ArXiv ID: 2312.01668 “View on arXiv”

Authors: Unknown

Abstract

This paper studies an optimal dividend problem with a drawdown constraint in a Brownian motion model, requiring the dividend payout rate to remain above a fixed proportion of its historical maximum. This leads to a path-dependent stochastic control problem, as the admissible control depends on its own past values. The associated Hamilton-Jacobi-Bellman (HJB) equation is a novel two-dimensional variational inequality with a gradient constraint, a type of problem previously only analyzed in the literature using viscosity solution techniques. In contrast, this paper employs delicate PDE methods to establish the existence of a strong solution. This stronger regularity allows us to explicitly characterize an optimal feedback control strategy, expressed in terms of two free boundaries and the running maximum surplus process. Furthermore, we derive key properties of the value function and the free boundaries, including boundedness and continuity. Numerical examples are provided to verify the theoretical results and to offer new financial insights.

Keywords: Stochastic Control, Hamilton-Jacobi-Bellman (HJB) Equation, Free Boundary Problem, Dividend Policy, Path-Dependent Constraints, Equities / Corporate Finance

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced PDE methods, variational inequalities, and free boundary analysis for a path-dependent stochastic control problem, indicating high mathematical complexity. While it includes numerical examples, the primary focus is on theoretical PDE existence and regularity proofs without reported backtests, live data, or implementation details for trading strategies.

  flowchart TD
    A["Research Goal:<br>Optimal Dividend Payout with Path-Dependent Drawdown Constraint"] --> B["Mathematical Modeling"]
    B --> C["Stochastic Control Problem<br>with Path-Dependent Constraint"]
    C --> D{"Analysis Method"}
    D -- Novel Approach --> E["PDE Methods<br>Strong Solution of HJB Equation"]
    D -- Previous Approach --> F["Viscosity Solution Techniques"]
    E --> G["Explicit Characterization<br>Free Boundaries & Feedback Control"]
    G --> H["Key Outcomes & Findings"]
    
    subgraph H [" "]
        I1["Optimal Dividend Strategy"]
        I2["Properties of Value Function"]
        I3["Numerical Verification"]
    end

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    style E fill:#f3e5f5
    style H fill:#e8f5e8
    style B fill:#fff3e0