# On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums

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## Abstract

This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximizing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton–Jacobi–Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber–Shiu functions. A number of concrete examples are analyzed.

## Keywords

Optimal strategy PDMP Barrier strategy Integro-differential HJB equation Gerber–Shiu function Stochastic controls## Mathematics Subject Classification

60G51 60G50 60K25 93E20## 1 Introduction

In classical collective risk theory, the surplus of an insurance company is described by the Cramér-Lundberg model. Under the assumption that the premium income per unit time is larger than the average amount claimed, the surplus in the Cramér-Lundberg model has positive first moment and has therefore the unrealistic property that it converges to infinity with probability one. In answer to this objection, De Finetti [1] introduced the dividend barrier model, in which all surpluses above a given level are transferred to a beneficiary, and raised the question of optimizing this barrier. In the mathematical finance and actuarial literature, there is a good deal of work being done on dividend barrier models and the problem of finding an optimal policy of paying dividends. Gerber and Shiu [2] and Jeanblanc and Shiryaev [3] consider the optimal dividend problem in a Brownian setting. Irbäck [4] and Zhou [5] study constant barriers. Asmussen et al. [6] investigate excess-of-loss reinsurance and dividend distribution policies in a diffusion setting. Azcue and Muler [7] take a viscosity approach to investigate optimal reinsurance and dividend policies in the Cramér-Lundberg model using a Hamilton–Jacobi–Bellman (HJB) system of equations. Avram et al. [8, 9], Kyprianou and Palmowski [11], Loeffen [12, 13], Loeffen and Renaud [14] and many other authors analyze the Lévy risk processes set up from the probabilistic point of view.

In this paper, we shall approach the dividend problem for a reserve-dependent risk process using the theory of piecewise deterministic Markov processes (PDMP). We also take into account the “severity” of ruin and therefore we consider the so-called Gerber–Shiu penalty function (see, e.g., Schmidli [15] or Avram et al. [9] and references therein). For this setup, without transaction costs, we find the corresponding HJB system. We analyze the barrier strategy for which all surpluses above a given level are transferred to dividends. In particular, we find necessary and sufficient conditions for the barrier strategy to be optimal.

We believe that PDMP models can better describe the situation of an insurance company, since, for example, they can invest the surplus into a bond with a fixed interest rate. Such a situation is described by a PDMP model with a linear premium (see [10]).

The paper is organized as follows. In Sect. 2, we introduce the basic notation and we describe the model we deal with. Section 3 is dedicated to the related one-sided and two-sided problems. In Sect. 4, we present the Verification Theorem, necessary and sufficient conditions for the barrier strategy to be optimal. In Sect. 6, we give all the proofs. Section 5 and 7 are devoted to some examples and concluding remarks.

## 2 The Model

### *Remark 2.1*

Constant and linear premium functions satisfy the above assumptions. For a constant premium function, we obtain the classical Cramér-Lundberg model.

## 3 Preliminaries

For the properties of the function \(G_{q,w}\), we refer the reader to [18], where numerous examples are studied.

## 4 Main Results

### **Theorem 4.1**

(Verification Theorem) Let \(\pi \) be an admissible dividend strategy such that \(v_\pi \) is absolutely continuous and ultimately dominated by some affine function. If (9) holds for \(v_\pi \) then \(v_\pi (x)=v(x)\) for all \(x\ge 0\).

The proof of all theorems given here will be given in Sect. 6.

### **Lemma 4.1**

Assume that the distribution function (d.f.) \(F\) of the claim size is absolutely continuous. Then the functions \(W_q\) and \(G_{q,w}\) are continuously differentiable for all \(x\ge 0\).

From now on, we assume that the claim size distribution is absolutely continuous with a density \(f\).

We will focus on the so-called barrier policy \(\pi _a\) transferring all surpluses above a given level \(a\) to dividends.

### **Theorem 4.2**

Finally, using the above two theorems, we can give necessary and sufficient conditions for the barrier strategy to be optimal.

### **Theorem 4.3**

### **Theorem 4.4**

### **Theorem 4.5**

Suppose that \(f\) is convex and \(p\) is concave. Then the barrier strategy at \(a^*\) is optimal, that is, \(v(x)=v_{a^*}(x)\) for all \(x\ge 0\).

### **Theorem 4.6**

## 5 Examples

Moreover, note that (14) is a Gerber–Shiu function with zero penalty function. In contrast to the one from (7), we now have \(\lim _{x\rightarrow \infty } W_q (x)=+\infty \). This means that the optimal value function under mild conditions is a linear combination of two Gerber–Shiu functions: An unstable one that vanishes on the negative half line and tends to infinity at infinity (corresponding to dividend payment, \(W_q\) in our notation), and a stable one, vanishing at infinity (corresponding to the penalty payment, \(G_{q,w}\) in our notation). From [18], we know that \(W_q\) equals the unstable solution of the fundamental system for (16). One can prove that there exists a unique unstable solution (see [18] for details). In the rest of this section, we will assume that the claim size has exponential distribution with intensity \(\mu \).

### 5.1 Linear Premium

Linear premium. Dependence of \(q\) on \(a^*\)

\(\mu =0.3\), \(\epsilon =0.02\), \(\lambda =0.1\), \(c=1\) | |||||
---|---|---|---|---|---|

\(q\) | 0.025 | 0.03 | 0.04 | 0.05 | 0.06 |

\(a^*\) | 17.82 | 13.42 | 8.42 | 5.33 | 3.18 |

Linear premium. Dependence of \(\mu \) on \(a^*\)

\(q=0.05\), \(\epsilon =0.02\), \(\lambda =0.1\), \(c=1\) | ||||||
---|---|---|---|---|---|---|

\(\mu \) | 0.25 | 0.3 | 0.4 | 0.5 | 0.6 | 1.1 |

\(a^*\) | 3.97 | 5.33 | 5.92 | 5.7 | 5.3 | 3.72 |

Linear premium. Dependence of \(\lambda \) on \(a^*\)

\(\mu =0.3\), \(q=0.05\), \(\epsilon =0.02\), \(c=1\) | |||||
---|---|---|---|---|---|

\(\lambda \) | 0.05 | 0.12 | 0.15 | 0.17 | 0.2 |

\(a^*\) | 4.84 | 5.03 | 4.08 | 3.1 | 1.07 |

### 5.2 Rational Premium

Rational premium. Dependence of \(q\) on \(a^*\)

\(\mu =0.3\), \(\lambda =0.1\), \(c=1\) | ||||
---|---|---|---|---|

\(q\) | 0.005 | 0.01 | 0.015 | 0.02 |

\(a^*\) | 37.03 | 23.98 | 17.16 | 12.77 |

Rational premium. Dependence of \(\mu \) on \(a^*\)

\(q=0.01\), \(\lambda =0.1\), \(c=1\) | ||||
---|---|---|---|---|

\(\mu \) | 0.15 | 0.2 | 0.25 | 0.3 |

\(a^*\) | 0 | 23.98 | 22.39 | 20.05 |

Rational premium. Dependence of \(\lambda \) on \(a^*\)

\(q=0.01\), \(\mu =0.3\), \(c=1\) | |||||
---|---|---|---|---|---|

\(\lambda \) | 0.05 | 0.12 | 0.15 | 0.2 | 0.25 |

\(a^*\) | 17.73 | 20.55 | 20.8 | 19.16 | 13.29 |

Note that \(a^*\) seems to have similar properties in both linear and rational premium examples.

## 6 Proofs

### 6.1 Proof of the Verification Theorem 4.1

The proof is based on a representation of \(v\) as the pointwise minimum of a class of “controlled” supersolutions of the HJB equation. We start with the observation that the value function satisfies a dynamic programming equation.

### **Lemma 6.1**

This follows by a straightforward adaptation of classical arguments (see, e.g., [7], pp. 276–277]). We will prove that \(v\) is a supersolution of the HJB equation.

### **Lemma 6.2**

### *Proof*

- (a)
\(V_s^\pi = W_s^\pi \), (b) \(W_s^\pi \ge \mathbb {E}[W^\pi _t|\mathcal {F}_s]\), where \(W^\pi \) is the process defined in (19).

### **Lemma 6.3**

We have \(v\in {\mathcal {G}}\).

### *Proof*

### **Proposition 6.1**

### *Proof*

### *Proof of the Verification Theorem 4.1*

### 6.2 Proof of Lemma 4.1

### 6.3 On the Value Function for the Barrier Strategy

### 6.4 Proofs of Necessary and Sufficient Conditions for Optimality of the Barrier Strategy

### *Proof of Theorem 4.3*

### *Proof of Theorem 4.4*

- i.
\((v'_{a^*}-v'_x)(x)=0\).

- ii.
\((v'_{a^*}-v'_x)(b)=W_q^\prime (b) \left( H_q^\prime (a^*)-H_q^\prime (x)\right) \ge 0\) for \(b\in [0,a^*]\) by the definition of \(a^*\).

- iii.
\((v'_{a^*}-v'_x)(u)=W_q^\prime (u)\left( H_q^\prime (u)-H_q^\prime (x)\right) \ge 0\) for \(u\in [a^*,x]\) by the assumption (13).

- iv.
\((v_{a^*}-v_x)(a^*)\ge 0\), thus by iii, \((v_{a^*}-v_x)(x)\ge 0\).

- v.
\((v_{a^*}-v_x)(x-z)\le (v_{a^*}-v_x)(x)\) for all \(z\ge 0\) by ii and iii.

Now assume that (12) does not hold. Then there exists \(x>a^*\) such that \((\mathcal {A}-q)v_{a^*}(x)>0 \). By the continuity of \((\mathcal {A}-q)v_{a^*}\), we deduce that \(\lim _{y\uparrow x}(\mathcal {A}-q)v_x(y)>0\), which contradicts (37). \(\square \)

### *Proof of Theorem 4.5*

Point (ii) is straightforward, and (iii) follows from the fact that \(g'(x)=0\) for any \(x< a^*\) and \(g\) is continuously differentiable. \(\square \)

### *Proof of Theorem 4.6*

## 7 Conclusions

In this paper, we solved the dividend problem with a penalty function at ruin. We found some sufficient and necessary conditions for a barrier strategy to be optimal. Unfortunately, some of them, like (12) and (13), may be difficult to verify. Moreover, we analyzed only single barrier strategies. Therefore, one can consider “multi-bands strategies” (see [9]). It would also be interesting to consider the effect of adding fixed transaction costs that have to be paid when dividends are being paid. In the next step, it would be reasonable to examine the so-called “dual model” with a negative premium function and positive jumps. In such a model, the premiums are regarded as costs and claims are viewed as profits. Such a model might be appropriate for a company that specializes in inventions and discoveries (see [20]). However, we leave these points for future research.

## Notes

### Acknowledgments

This work is partially supported by the National Science Centre under the grant DEC-2013/09/B/ST1/01778. The second author kindly acknowledges partial support by the project RARE -318984, a Marie Curie IRSES Fellowship within the 7th European Community Framework Programme.

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