MV Singular Dividend Control

• The paper introduces a mean–variance dividend control model that incorporates a variance penalty, creating a time-inconsistent framework resolved using equilibrium strategies.
• It formulates the surplus process via drifted Brownian motion and derives explicit analytical and numerical solutions for both high and low risk-aversion regimes.
• The study establishes an extended verification theorem within an HJB-variational system, emphasizing barrier strategies and the critical role of risk aversion in dividend optimization.

The MV singular dividend control problem refers to the singular stochastic control problem in which one seeks to maximize the mean-variance (MV) functional of discounted dividend payments until ruin, for a surplus process governed by a drifted Brownian motion subject to singular dividend payout controls. This formulation extends the classical de Finetti optimal dividend problem by introducing aversion to variability in dividend payouts, and thereby incorporates a risk/variance penalty in the objective. Unlike its classical counterpart, the MV criterion renders the problem time-inconsistent, requiring equilibrium (rather than optimal) strategies established via an intrapersonal game-theoretic framework. The mathematical mechanisms, equilibrium classification, verification theorem, and numerical evaluation are described below.

1. Model Formulation and Surplus Dynamics

The MV singular dividend control problem is set on a filtered probability space $$(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, \mathbb{P})$$ supporting a standard Brownian motion $B$. The uncontrolled surplus process evolves according to

$dX_t = a\,dt + b\,dB_t,$

for fixed drift $a>0$ and volatility $b>0$. The dividend control is represented by a nondecreasing, $\mathcal{F}_t$-adapted, càdlàg process $D = (D_t)_{t \geq 0}$ with the technical constraint $\Delta D_t \leq X_{t^-}$ to ensure that dividend payments do not exceed available surplus.

Under an admissible strategy $D$, the controlled surplus is given by the singular stochastic differential equation (SDE)

$$dX_t^D = a\,dt + b\,dB_t - dD_t,\quad X_{0^-} = x > 0,$$

with ruin time defined by

$\tau^D = \inf\{ t \geq 0 : X_t^D \leq 0 \}.$

Dividend payments are frozen upon ruin: $D_t = D_{\tau^D}$ for all $t \geq \tau^D$.

2. Mean-Variance Dividend Objective

Let $\rho > 0$ denote the discount rate. The total discounted future dividends, for $t < \tau^D$, are prescribed by

$Y_t^D = \int_t^{\tau^D} e^{-\rho(s-t)}\,dD_s,$

with $Y_t^D = 0$ for $t \geq \tau^D$. The MV criterion—incorporating both expected value and a variance penalty—is parametrized by a risk-aversion coefficient $\gamma > 0$: $$J(x, t; D) = \mathbb{E}_{x, t}[Y_t^D] - \frac{\gamma}{2} \operatorname{Var}_{x, t}[Y_t^D],$$ where the expectation and variance are conditional on $X_{t^-}^D = x$. Equivalently, letting $G(x, t) = \mathbb{E}_{x, t}[Y_t^D]$ and $H(x, t) = \mathbb{E}_{x, t}[(Y_t^D)^2]$, the criterion can be expressed as

$$J(x, t; D) = G(x, t) - \frac{\gamma}{2}(H(x, t) - G(x, t)^2).$$

This objective balances maximizing the expected dividend payouts with minimizing their variability before ruin.

3. Time-Inconsistency and Game-Theoretic Equilibrium Definition

The inclusion of a variance penalty in the objective destroys the Bellman optimality principle, making the control problem dynamically inconsistent—an optimal policy at $t$ need not remain optimal at future times $s > t$. To address this, the problem is reframed as an intrapersonal dynamic game: strategies must constitute weak equilibria in the sense of Björk–Murgoci (2010) and Dai–Jin–Jia (2024). The equilibrium requirement is as follows: given a candidate strategy $D^*$, perturbed over a small interval $[t, t + \varepsilon)$ by lump-sum payout $d \leq x$, the first-order change in $J$ must be nonpositive as $\varepsilon \to 0$: $$\liminf_{\varepsilon \to 0^+} \frac{J(x, t; D^*) - J(x, t; D^\varepsilon)}{\varepsilon} \geq 0,$$ for every $x > 0$, $t \geq 0$, $d \in [0, x]$, where $D^\varepsilon$ is the perturbed strategy. The equilibrium dividend strategy $D^*$ yields the equilibrium value function $V(x, t) = J(x, t; D^*)$.

4. Verification Theorem and HJB-Variational System

The extended verification theorem is central to the solution methodology. Define the second-order differential operator

$$\mathcal{M}\phi(x, t) = \partial_t \phi + a\,\partial_x \phi + \frac{1}{2}b^2\,\partial_{xx}^2 \phi.$$

The state-time space $\mathbb{R}_+^2$ is decomposed into “pay” and “no-transaction” regions based on the gradient of the candidate value function: $$\operatorname{Pay} := \{ (x, t) : \partial_x V(x, t) = 1 \}, \qquad \operatorname{NT} := \mathbb{R}_+^2 \setminus \operatorname{Pay}.$$ Theorem 3.1 stipulates that if smooth functions $V, G, H$ satisfy: (i) boundary conditions $V(0, t) = G(0, t) = H(0, t) = 0$; (ii) the extended HJB-variational system

$$\begin{cases} \max\left\{ \mathcal{M}V - \frac{\gamma}{2} \mathcal{M}(G^2) + \gamma G\, \mathcal{M}G - \rho G + \gamma\rho (H - G^2),\; 1 - \partial_x V \right\} = 0,\ (\mathcal{M}G - \rho G)\,1_{\operatorname{NT}} + (1 - \partial_x G)\,1_{\operatorname{Pay}} = 0,\ (\mathcal{M}H - 2\rho H)\,1_{\operatorname{NT}} + (2G - \partial_x H)\,1_{\operatorname{Pay}} = 0, \end{cases}$$

then the minimal reflection strategy (lump-sum payout in $\operatorname{Pay}$, local-time reflection at the region boundary) is a time-consistent equilibrium. The equilibrium $V(x, t)$ is realized, with

$$G(x, t) = \mathbb{E}_{x, t}[Y_t^{D^*}],\quad H(x, t) = \mathbb{E}_{x, t}[(Y_t^{D^*})^2].$$

This verification theorem enables constructive identification of equilibrium solutions in analytic and semi-explicit form under suitable smoothness and concavity conditions.

5. Analytical Equilibria: Large and Small Risk Aversion

The paper characterizes two principal equilibrium regimes depending on the risk aversion parameter $\gamma$:

Case A: High Risk Aversion ($\gamma \geq 2a/b^2$)

The unique equilibrium is immediate liquidation: all surplus is paid out at $t = 0$, triggering instant ruin. Explicitly,

$G(x, t) = x,\quad H(x, t) = x^2,\quad V(x, t) = x.$

The value function is linear in surplus $x$; the criterion’s gradient always meets the payout region condition $\partial_x V = 1$.

Case B: Low Risk Aversion ($0 < \gamma < \varepsilon \leq 2a/b^2$)

Here, the equilibrium takes the form of a constant-barrier strategy: pay all surplus down to barrier $\tilde{x}$ when above it, otherwise wait. On the no-pay region $[0, \tilde{x})$, the relevant ODEs are

$$- \rho G + a G' + \frac{1}{2}b^2 G'' = 0;\quad -2\rho H + a H' + \frac{1}{2}b^2 H'' = 0,$$

with solutions

$$G(x) = C_1(e^{r_1 x} - e^{r_2 x}),\quad H(x) = C_3(e^{r_3 x} - e^{r_4 x}),$$

where

$$r_{1,2} = \frac{-a \pm \sqrt{a^2 + 2\rho b^2}}{b^2},\quad r_{3,4} = \frac{-a \pm \sqrt{a^2 + 4\rho b^2}}{b^2}.$$

On the pay region $[ \tilde{x}, \infty )$, continuity and matching requirements yield

$$G(x) = G(\tilde{x}) + (x - \tilde{x}),\quad H(x) = H(\tilde{x}) + 2G(\tilde{x})(x - \tilde{x}) + (x - \tilde{x})^2.$$

Smooth pasting fixes the constants, and the barrier $\tilde{x}$ solves an explicitly derived nonlinear equation $f(\tilde{x}, \gamma) = 0$, with concavity requirement $V''(x) < 0$ domainwise.

6. Numerical Illustration and Parametric Regimes

For parameters $a = 1$, $b = 0.25$, $\rho = 0.2$, the critical risk aversion is numerically $\frac{2a}{b^2} = 32$. For $\gamma = 0$, the classical de Finetti result is recovered with barrier $\tilde{x} \approx 0.3141$. The function $f(x, \gamma)$ admits a unique positive root $\tilde{x}(\gamma)$ increasing with $\gamma$ up to approximately $\bar{\gamma} \approx 0.1397 \ll 32$, beyond which concavity fails. For instance, at $\gamma = 0.13$, the barrier is $\tilde{x}(0.13) \approx 0.3232$, and the corresponding value function is strictly concave below the barrier.

Key parametric findings include:

• The equilibrium barrier $\tilde{x}(\gamma)$ is strictly increasing in $\gamma$ for $\gamma \in [0, \bar{\gamma}]$.
• The equilibrium payoff $V(x; \gamma)$ decreases with increasing $\gamma$, as expected from stronger penalties on variance.
• Increasing impatience ($\rho$) lowers the barrier, while higher drift ($a$) or reduced volatility ($b$) modify admissible risk aversion and the barrier in line with economic intuition.

The MV singular dividend control problem thus presents a distinct dichotomy: high risk aversion leads to immediate payout (liquidation), while lower risk tolerance generates an explicit barrier equilibrium up to a numerically determined threshold for $\gamma$. This threshold is strictly less than $2a/b^2$ except in the degenerate liquidation regime.

7. Relation to Classical and Multidimensional Dividend Control

The MV singular dividend problem generalizes the de Finetti classical dividend control by integrating a reward-risk tradeoff directly into the objective. The time-inconsistency induced by the MV criterion necessitates equilibrium methods not present in the classical Bellman HJB approach. This framework is conceptually distinct from multidimensional singular dividend/switching problems such as those treated in Azcue–Muler (Azcue et al., 2018), which concern n-dimensional compound Poisson surplus models with optimal switching and penalty at ruin, addressed via viscosity solution theory and monotone finite-difference fixed-point schemes. Nonetheless, both lines of research rigorously elaborate the interplay between singular controls, dynamic programming, and non-standard objectives, with the MV singular control problem providing a mean–variance-driven paradigm for solvency and payout optimization.