Optimality of refraction strategies for a constrained dividend problem

TL;DR

This work establishes that, under absolutely continuous dividend strategies with a ruin-time constraint, optimal control in both spectrally negative and spectrally positive Lévy models is realized by threshold (refraction) strategies, with the dividend rate active only above a critical level. The authors develop a dual Lagrangian framework, derive explicit expressions for the value functions and ruin indices via scale functions, and characterize the optimal Lagrange multiplier through complementary slackness, providing a complete solution to the constrained problem. They extend known SN/SP results to the constrained setting, verify optimality through smoothness and HJB checks, and illustrate the theory with numerical experiments that highlight how the optimal barrier and the Lagrange multiplier vary with model parameters. The results offer a tractable, fluctuation-identity–driven approach to realistic dividend policies under longevity constraints, with clear guidance for implementation in actuarial and risk-management contexts.

Abstract

We consider de Finetti's problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

Figures (7)

• Figure 1: Plots $b \mapsto \xi_\Lambda(b)$ for Case 1 (left) and Case 2 (right). The points at $b_\Lambda$ are indicated by squares.
• Figure 2: Plots of $x \mapsto V_\Lambda(x)$ (solid) for Case 1 (left) and Case 2 (right). Suboptimal value functions $v_\Lambda^{b}$ (dotted) are also plotted for the choice of $b = 0, \bar{b}_\Lambda/2, 3 \bar{b}_\Lambda/2$ for Case 1 and $b = 2,4,6$ for Case 2. The points at $b_\Lambda$ are indicated by squares and those at $b$ in the suboptimal cases are indicated by up- (resp. down-) pointing triangles when $b > b_\Lambda$ (resp. $b < b_\Lambda$).
• Figure 3: (Left) Plots of $x \mapsto V_\Lambda(x; K)$ for $\Lambda = 0.1$, $\ldots$, $1$, $2$, $\ldots$, $10$, $20$, $\ldots$, $100$, $200$, $\ldots$, $1000$, $2000$, $\ldots$, $10000$, $20000$ (dotted) and for $\Lambda = 0$ (solid, bold face) for the case $K = 0.1$. The two vertical dotted lines indicate the values of $\underline{x}$ and $\overline{x}$ such that $K_{\underline{x}} = K$ and $\Psi_{\overline{x}}(b_0) = K$. On $[\underline{x}, \overline{x}]$, the minimum of $V_\Lambda(x; K)$ over $\Lambda$ is shown in solid fold-face red line. (Right) Plots of the Lagrange multiplier $\Lambda^*$ on $(\underline{x}, \overline{x}]$ with the same two vertical lines as in the left plot.
• Figure 4: Plots of $V(x; K)$ (left) and the Lagrange multiplier $\Lambda^*$ (right) as functions of $x$ and $K$.
• Figure 5: Plots of $x \mapsto \overline{V}_\Lambda(x)$ along with suboptimal value functions $\bar{v}_\Lambda^{b}$ (dotted) for the choice of $b = 0, \bar{b}_\Lambda/2, 3 \bar{b}_\Lambda/2$. The point at $\bar{b}_\Lambda$ is indicated by a square and those at $b$ in the suboptimal cases are indicated by up- (resp. down-) pointing triangles when $b > \bar{b}_\Lambda$ (resp. $b < \bar{b}_\Lambda$).

Theorems & Definitions (39)

• Remark 4.2
• Remark 4.4
• Proposition 4.6
• Remark 4.7
• Lemma 4.8
• Remark 4.9
• Remark 4.10
• Lemma 4.11
• proof
• Proposition 4.12