Stochastic Control of Dividends with a Drawdown Penalty

TL;DR

This paper analyzes a diffusion surplus model with dividends paid at a bounded rate and a drawdown penalty, formulating a stochastic control problem to maximize discounted dividends minus time in drawdown beyond a threshold. By deriving and solving a Hamilton–Jacobi–Bellman equation, the authors show the optimal dividend policy is bang-bang, taking either $0$ or the maximal rate $u_0$. They identify a critical value $\zeta$ that delineates regimes: for $\beta\ge\zeta$ the optimal strategy is to pay dividends at the maximal rate continuously, while for $\beta<\zeta$ a drawdown-sensitive regime with switching boundaries $z_f$ and $z_g$ governs the optimal policy, and the value function is constructed via a two-subproblem decomposition with a crossing constant $C$. The results provide an explicit, piecewise-defined $v(z)$ and illuminate how the drawdown penalty alters dividend strategies in diffusion settings, complemented by numerical illustrations and discussion of practical implications.

Abstract

We consider a diffusion risk model where dividends are paid at rate $U(t) \in [0, u_0]$. We are interested in maximising the dividend payments under a drawdown constraint, that is, we penalise a drawdown size larger than a level $d > 0$. We show that the optimal dividend rate $U(t)$ is either zero or the maximal rate $u_0$ and determine the optimal strategy. Moreover, we derive an explicit expression for the value function by solving a system of differential equations.

Theorems & Definitions (2)

• Lemma 1.1
• proof