Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal dividend payout with path-dependent drawdown constraint

Chonghu Guan, Jiacheng Fan, Zuo Quan Xu

TL;DR

This work addresses optimal dividend payout under a path-dependent drawdown constraint in a Brownian surplus model, where the payout rate must stay above a fixed fraction of its running maximum. The authors develop a new PDE-based approach to a two-dimensional Hamilton–Jacobi–Bellman variational inequality with a gradient constraint, enabling the derivation of strong (ceiling model) and weak (no-ceiling model) solutions and yielding explicit, bang-bang feedback controls defined by two free boundaries. They prove existence, uniqueness, and regularity of solutions, characterize the switching and converting boundaries, and provide a comprehensive numerical study that validates the theory and offers financial intuition under various parameter regimes. The framework extends prior viscosity-solution results to stronger solution concepts, derives explicit optimal strategies, and addresses a challenging path-dependent control problem with potential applicability to jump-diffusion extensions.

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.

Optimal dividend payout with path-dependent drawdown constraint

TL;DR

This work addresses optimal dividend payout under a path-dependent drawdown constraint in a Brownian surplus model, where the payout rate must stay above a fixed fraction of its running maximum. The authors develop a new PDE-based approach to a two-dimensional Hamilton–Jacobi–Bellman variational inequality with a gradient constraint, enabling the derivation of strong (ceiling model) and weak (no-ceiling model) solutions and yielding explicit, bang-bang feedback controls defined by two free boundaries. They prove existence, uniqueness, and regularity of solutions, characterize the switching and converting boundaries, and provide a comprehensive numerical study that validates the theory and offers financial intuition under various parameter regimes. The framework extends prior viscosity-solution results to stronger solution concepts, derives explicit optimal strategies, and addresses a challenging path-dependent control problem with potential applicability to jump-diffusion extensions.

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.
Paper Structure (24 sections, 40 theorems, 267 equations, 13 figures)

This paper contains 24 sections, 40 theorems, 267 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Model formulation
  3. The ceiling model: $\overline{c}<\infty$
  4. The boundary case: $V(x,\overline{c})$
  5. The non-boundary case: $V(x,c)$
  6. The no-ceiling model: $\overline{c}=+\infty$
  7. The value function and HJB equation
  8. Numerical study
  9. Optimal dividend payout strategies
  10. Comparative statics
  11. Concluding remarks
  12. Proofs of Proposition \ref{['prop:boundary']}, Lemma \ref{['lem:g']}, and Theorem \ref{['op2']}
  13. Proof of Proposition \ref{['prop:boundary']}
  14. Proof of Lemma \ref{['lem:g']}
  15. Proof of Theorem \ref{['op2']}
  16. ...and 9 more sections

Key Result

Proposition 3.1

Suppose $g\in C^2({\mathbb R}^+)$ satisfies the following ODE: If both $g$ and $g'$ are bounded, then $g$ is the value function of the problem boundary, i.e., Moreover, an optimal dividend payout strategy is given by $\{\mathcal{C}^*_{t}\}_{t\geqslant0}$, where

Figures (13)

  • Figure 1: $2\mu \overline{c} < \sigma^2 r$, $\mu=0.1$, $r=0.08$, $\sigma=0.8$ and $\bar{c}=0.1$
  • Figure 2: $2\mu \overline{c} >\sigma^2 r$, $\mu=0.4$, $r=0.05$, $\sigma=0.4$ and $\bar{c}=0.3$ and $b=1$
  • Figure 3: The value functions $c\mapsto v(x_{i},c)$ are in solid curve, from below to top corresponding to $x_{1}<x_{2}<x_{3}<x_{4}$. The dashed curve is $\mathscr{X}(\cdot)$. Its blue parts stand for $\mathfrak{m}(x,c_{1})$ and $\mathfrak{m}(x,c_{2})$ at all different levels of $x$, whereas the red parts stand for $\mathfrak{M}(x,c_{1})$ and $\mathfrak{M}(x,c_{2})$. If a point belongs to ${\cal S}$ (resp. ${\cal NS}$), then so does any point above (resp. below) it. Hence, all points above (resp. below) dashed curve belong to ${\cal S}$ (resp. ${\cal NS}$). The blue (resp. red) parts of the dashed curve belong to ${\cal S}$ (resp. ${\cal NS}$). Therefore, the blue parts of the solid curve are expanding as $x$ gets bigger and eventually the whole curve becomes blue above certain threshold (which is indeed $\sup_{c\in[0,\overline{c})}\mathscr{X}(c)$). Although the dashed curve in above demonstration figure is $W$-shaped, we believe it should be increasing as $c$ gets bigger. Unfortunately, we cannot prove this, but numerical examples in Section \ref{['sec:numerical']} strongly support our conjecture.
  • Figure 4: Comparison of numerical solutions and cases with no constraints for $b=0$ and $b=1$.
  • Figure 5: Illustration of optimal trajectories for different starting points $(x,c)$ across regions. The arrows indicate the evolution paths from initial states in different domains. Parameters: $\mu=0.3$, $b=0.6$, $r=0.05$, $\sigma=0.3$, $\bar{c}=0.3$.
  • ...and 8 more figures

Theorems & Definitions (42)

  • Proposition 3.1: Verification for the boundary case
  • Lemma 3.2
  • Theorem 3.3: Solution for the boundary case
  • Theorem 3.4: Solution for the simple case $2\mu\overline{c} \leqslant \sigma^2 r$
  • Definition 3.1: Strong solution
  • Theorem 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Proposition 3.8
  • Theorem 3.9: Optimal value and optimal strategy in the complicated case.
  • ...and 32 more