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Equilibrium Strategies for Singular Dividend Control Problems under the Mean-Variance Criterion

Jingyi Cao, Dongchen Li, Virginia R. Young, Bin Zou

TL;DR

This paper addresses a mean-variance (MV) mean-variance dividend problem under a singular control framework with dividends paid up to the endogenous ruin time $\tau$. It develops a new verification theorem for MV with the integral objective $Y_t = \int_t^\tau e^{-\rho(s-t)} dD_s$, yielding a three-function extended HJB system with $V$, $G$, and $H$, and characterizes time-consistent equilibrium strategies via a pay region $\mathrm{P}$ and a no-transaction region $\mathrm{NT}$ coupled to a Skorokhod reflection problem. The authors obtain two main equilibria: (i) for large risk aversion $\gamma \ge \dfrac{2a}{b^2}$, pay all surplus immediately; (ii) for small $\gamma$, a time-independent barrier strategy with barrier $\tilde{x}$ and explicit forms for $G$, $H$, and $V$ in the two regions, verified under a concavity condition. Numerical examples illustrate the barrier behavior, show a threshold $\bar{\gamma}$ below which the barrier is viable, and reveal open questions for intermediate $\gamma$. The results contribute a novel MV-singular-control framework for dividends with endogenously determined stopping times and connect to existing literature on time-inconsistent control and equilibrium strategies.

Abstract

We revisit the optimal dividend problem of de Finetti by adding a variance term to the usual criterion of maximizing the expected discounted dividends paid until ruin, in a singular control framework. Investors do not like variability in their dividend distribution, and the mean-variance (MV) criterion balances the desire for large expected dividend payments with small variability in those payments. The resulting MV singular dividend control problem is time-inconsistent, and we follow a game-theoretic approach to find a time-consistent equilibrium strategy. Our main contribution is a new verification theorem for the novel dividend problem, in which the MV criterion is applied to an integral of the control until ruin, a random time that is endogenous to the problem. We demonstrate the use of the verification theorem in two cases for which we obtain the equilibrium dividend strategy (semi-)explicitly, and we provide a numerical example to illustrate our results.

Equilibrium Strategies for Singular Dividend Control Problems under the Mean-Variance Criterion

TL;DR

This paper addresses a mean-variance (MV) mean-variance dividend problem under a singular control framework with dividends paid up to the endogenous ruin time . It develops a new verification theorem for MV with the integral objective , yielding a three-function extended HJB system with , , and , and characterizes time-consistent equilibrium strategies via a pay region and a no-transaction region coupled to a Skorokhod reflection problem. The authors obtain two main equilibria: (i) for large risk aversion , pay all surplus immediately; (ii) for small , a time-independent barrier strategy with barrier and explicit forms for , , and in the two regions, verified under a concavity condition. Numerical examples illustrate the barrier behavior, show a threshold below which the barrier is viable, and reveal open questions for intermediate . The results contribute a novel MV-singular-control framework for dividends with endogenously determined stopping times and connect to existing literature on time-inconsistent control and equilibrium strategies.

Abstract

We revisit the optimal dividend problem of de Finetti by adding a variance term to the usual criterion of maximizing the expected discounted dividends paid until ruin, in a singular control framework. Investors do not like variability in their dividend distribution, and the mean-variance (MV) criterion balances the desire for large expected dividend payments with small variability in those payments. The resulting MV singular dividend control problem is time-inconsistent, and we follow a game-theoretic approach to find a time-consistent equilibrium strategy. Our main contribution is a new verification theorem for the novel dividend problem, in which the MV criterion is applied to an integral of the control until ruin, a random time that is endogenous to the problem. We demonstrate the use of the verification theorem in two cases for which we obtain the equilibrium dividend strategy (semi-)explicitly, and we provide a numerical example to illustrate our results.

Paper Structure

This paper contains 6 sections, 3 theorems, 44 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Verification theorem
  4. Equilibrium dividend strategies
  5. Numerical examples
  6. Conclusions

Key Result

Theorem 3.1

Let $\widetilde{V}$, $G$, and $H$ be three functions, all mapping from $(x, t) \in \mathbb{R}_+^2$ to $\mathbb{R}$. Define the pay region $\mathrm{P}$ and no-transaction region $\mathrm{NT}$, respectively, by Suppose that $\widetilde{V}$, $G$, and $H$ satisfy the following conditions: Then, $\widetilde{V}$ is an equilibrium value function defined in eq:V, and $D^*$ is a time-consistent equilibri

Figures (7)

  • Figure 1: $f(x, 0.15)$ in \ref{['eq:xt']} (left) and $V"(x)$ defined via \ref{['eq:Vx']} (right) when $\gamma = 0.15$
  • Figure 2: $f(x, 40)$ in \ref{['eq:xt']} (left) and the corresponding $V$s in \ref{['eq:Vx']} when $\gamma = 40$
  • Figure 3: The barrier $\tilde{x}$ as a function of $\gamma$ (left) and $f(x, 0.13)$ in \ref{['eq:xt']} when $\gamma = 0.13$ (right)
  • Figure 4: The value function $V(x)$ (left) and its zoom-in for $x \in (0.1, 0.4)$ (right)
  • Figure 5: Impact of the discount rate $\rho$ on the maximum allowed risk aversion $\bar{\gamma}$ (left) and the barrier $\tilde{x}$ under $\gamma = 0.1396$ (right)
  • ...and 2 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 3 more