Optimal dividend payout with path-dependent drawdown constraint

Paper: arXiv 2312.01668

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.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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

  style A fill:#e1f5fe
  style E fill:#f3e5f5
  style H fill:#e8f5e8
  style B fill:#fff3e0

Cited References (29)

1. Optimal Ratcheting of Dividend Payout Under Brownian Motion Surplus — → 2308.15048
2. Optimal dividends under a drawdown constraint and a curious square-root rule (2022) — link
3. Optimal Investment and Consumption under a Habit-Formation Constraint (2021) — link
4. Optimal Ratcheting of Dividends in a Brownian Risk Model (2020) — link
5. Optimal consumption with reference to past spending maximum (2020) — link
6. Optimal Ratcheting of Dividends in Insurance (2019) — link
7. Portfolio selection with consumption ratcheting (2018) — link
8. Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates (2018) — link

1 of 29 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.