Optimal dividend payout with path-dependent drawdown constraint

TL;DR

This paper addresses an optimal dividend payout problem with a path-dependent drawdown constraint, employing PDE methods to characterize the optimal control and analyze the value function in a stochastic model.

Contribution

It introduces a novel PDE approach to solve a path-dependent stochastic control problem with a drawdown constraint, providing explicit control strategies and theoretical properties.

Findings

Explicit optimal feedback control strategy derived

Existence of a strong solution to the PDE established

Numerical examples verify theoretical results and offer financial insights

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.