[Paper Review] On Gerber-Shiu functions and optimal dividend distribution for a Lévy risk-process in the presence of a penalty function

Florin Avram, Zbigniew Palmowski+1 more|arXiv (Cornell University)|Oct 22, 2011

Probability and Risk Models48 references45 citations

TL;DR

This paper solves the optimal dividend distribution problem for a spectrally negative Lévy risk process with a penalty function at ruin and fixed transaction costs. It establishes that the value function is the unique stochastic solution to the Hamilton-Jacobi-Bellman (HJB) equation and identifies a necessary and sufficient condition for optimality of a single dividend-band strategy using a Gerber–Shiu function, providing a complete analytical solution for this class of stochastic control problems in insurance risk theory.

ABSTRACT

This paper concerns an optimal dividend distribution problem for an insurance company which risk process evolves as a spectrally negative L\'{e}vy process (in the absence of dividend payments). The management of the company is assumed to control timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividend payments received until the moment of ruin and a penalty payment at the moment of ruin which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. A complete solution is presented to the corresponding stochastic control problem. It is established that the value-function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated HJB equation. Furthermore, a necessary and sufficient condition is identified for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. A number of concrete examples are analyzed.

Motivation & Objective

• To solve the optimal dividend distribution problem for an insurance company whose surplus follows a spectrally negative Lévy process.
• To incorporate a penalty function at ruin that depends on the size of the shortfall, reflecting financial consequences of insolvency.
• To account for fixed costs associated with dividend payments, modeling real-world transaction frictions.
• To identify conditions under which a single dividend-band strategy is optimal, using the Gerber–Shiu function as a key analytical tool.
• To establish the value function as the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated Hamilton-Jacobi-Bellman (HJB) equation.

Proposed method

• Formulates the problem as a singular stochastic control problem with absorption at ruin and a penalty function on the deficit.
• Derives the Hamilton-Jacobi-Bellman (HJB) equation governing the optimal value function, involving integro-differential operators based on the Lévy generator.
• Uses the scale function $W^{(q)}$ associated with the spectrally negative Lévy process to express the value function and derive explicit solutions.
• Introduces the Gerber–Shiu function $G(b^-, y)$ as a key analytical tool to characterize the optimality of dividend-band strategies.
• Applies verification theorems to prove that the value function is the minimal stochastic supersolution of the HJB equation.
• Employs Laplace transforms and Fubini’s theorem to analyze the monotonicity and optimality conditions of the control policy.

Experimental results

Research questions

• RQ1 Under what conditions is a single dividend-band strategy optimal for a spectrally negative Lévy risk process with a penalty at ruin?
• RQ2 How does the inclusion of fixed transaction costs affect the structure of the optimal dividend strategy?
• RQ3 What is the role of the Gerber–Shiu function in characterizing the optimality of dividend-band strategies?
• RQ4 How does the value function relate to the scale function $W^{(q)}$ and the Lévy process characteristics?
• RQ5 Can the value function be characterized as the unique stochastic solution and the smallest stochastic supersolution of the HJB equation?

Key findings

• The value function is the unique stochastic solution and the pointwise smallest stochastic supersolution of the associated Hamilton-Jacobi-Bellman equation.
• A necessary and sufficient condition for the optimality of a single dividend-band strategy is expressed in terms of the Gerber–Shiu function $G(b^-, y)$, specifically that $G(b^-, b^+ + c) < G(b^-, b^+)$ for all $c > 0$.
• The optimality condition is equivalent to the ultimate monotonicity of the Gerber–Shiu function $G(b^-, y)$ and its transformed version $G^\#(y)$.
• For the case with no fixed costs ($K=0$), the optimality condition is derived via Laplace transforms, showing that $\mathcal{L}g(\theta) \cdot \frac{\theta}{\psi(\theta) - q} = \frac{e^{\theta b^+}}{\psi(\theta) - q} \int_{[b^+,\infty)} e^{-\theta z} Z^{(q,\theta)'}(z) G(dz)$, which is completely monotone.
• The continuity of the value function is established via a verification argument involving the limit of $f_\varepsilon(x,y)$ as $|x-y| \to 0$, using the strong Markov property and dominated convergence.
• In the bounded variation case, the second factor in the estimate $f_\varepsilon(x,y) \leq \text{sup}_{x\in[0,\delta]} V_w(x) - w(-\delta) \cdot E^y[e^{-q\tau_\delta^\pi} \mathbf{1}_{\{\tau_\delta^\pi < \tau_0^\pi\}}]$ tends to zero as $\delta \to 0$, ensuring continuity of the value function.