##
**Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model.**
*(English)*
Zbl 1136.91016

The paper deals with the problem of optimal dynamic risk control and dividends distribution of a financial corporation with the reserve \((X_t)_{t\geq 0}\) described by a Cramér-Lundberg process. The risk of the corporation is controlled by reinsurance policy. The reinsurance policy is a Borel measurable function \(R:[0,\infty)\rightarrow[0,\infty)\) such that \(0\leq R(\alpha)\leq \alpha\), where \(R(\alpha)\) is the part of the claim, that the company pays, when the size of the claim is \(\alpha\).

The optimal control problem can be formulated as follows. Let \(\Omega\) be a set of paths with left and right limits and let the process of reserve \((X_t)_{t\geq 0}\) be defined on the complete probability space \((\Omega,{\mathcal F},({\mathcal F}_t)_{t\geq 0},\mathbf{P})\). Denote \(\mathcal R\) a family of reinsurance polices. Define a set of admissible control strategies \(\Pi=\{\pi: \pi=(R_t,L_t)\}\), where \(R_t\in {\mathcal R}\) and \(L_t\) is the cumulative amount of dividends paid out up to time \(t\). Given an admissible control strategy \(\pi\), the control risk process \(X_t^{\pi}\) is given by \(X_t^{\pi}=x+\int_0^t p_{R_t}d s-\sum_{i=1}^{N_t}R_{\tau_i}(U_i)-L_t\), where \(\tau_i\) is the time of occurrence of the \(i\)th claim. The corresponding ruin time of the company \(\tau^\pi=\inf\{t\geq 0:X_t^\pi<0\}\). The return function is \(V_\pi(x)=\mathbf{E}_x\left(\int_0^{\tau^\pi}\exp{(-cs)}d L_s\right)\), where \(c\) is discount factor. The optimal return function is defined as \(V(x)=\sup_{\pi\in\Pi}V_\pi(x)\).

In this paper the optimal return function \(V(x)\) is characterized as the smallest of the viscosity solutions of the associated HJB equation [see M. G. Crandall and P.-L. Lions, Trans. Am. Math. Soc. 277, 1–42 (1983; Zbl 0599.35024)]. It is proved, based on this characterisation of the optimal return function \(V(x)\), that there exists an optimal admissible strategy \(\pi^\star\) and this is a stationary strategy i.e. the decision of which reinsurance policy to choose and how much to pay out as dividend depends only on the current reserve.

The optimal control problem for restricted classes of reinsurance policies are also considered. The following families of reinsurance policies are defined: \({\mathcal R}_0\) the case of no reinsurance; \({\mathcal R}_P\) the set of proportional reinsurance policies; \({\mathcal R}_{XL}\) the case of excess-of -loss reinsurance policies and \({\mathcal R}_A\) the set of all the reinsurance policies. The optimal reinsurance policy in each of the families \({\mathcal R}_P\) [cf. H. Schmidli, Scand. Actuarial J. 2001, No. 1, 55–68 (2001; Zbl 0971.91039)], \({\mathcal R}_{XL}\), and \({\mathcal R}_{A}\) is described and it is proved that the optimal dividend payment policy can be of three possible form: the incoming premium is paid out directly as dividends, no dividend is paid, or a positive amount is paid out immediately.

In the case of no reinsurance, H. Gerber [Mitt. Verein. Schweiz. Versicherungsmath. 69, 185–228 (1969; Zbl 0193.20501)] has investigated the optimal return function [see also H. Bühlmann, Mathematical methods in risk theory. Springer Verlag, Berlin (1970; Zbl 0209.23302)]. The related problems of optimal dividend payout, when the reserve process is modelled as Brownian motion, has been solved recently by S. Asmussen and M. Taksar [Insur. Math. Econ. 20, No. 1, 1–15 (1997; Zbl 1065.91529)] in the case of no reinsurance, B. Højgaard and M. Taksar [Math. Finance 9, No. 2, 153–182 (1999; Zbl 0999.91052)] in the case of proportional reinsurance, and S. Asmussen, B. Højgaard and M. Taksar [Finance Stoch. 4, No. 3, 299–324 (2000; Zbl 0958.91026)] and T. M. Choulli, M. Taksar and X. Y. Zhou [Quant. Finance 1, 573–596 (2001)] in the case of excess-of-loss reinsurance.

The optimal control problem can be formulated as follows. Let \(\Omega\) be a set of paths with left and right limits and let the process of reserve \((X_t)_{t\geq 0}\) be defined on the complete probability space \((\Omega,{\mathcal F},({\mathcal F}_t)_{t\geq 0},\mathbf{P})\). Denote \(\mathcal R\) a family of reinsurance polices. Define a set of admissible control strategies \(\Pi=\{\pi: \pi=(R_t,L_t)\}\), where \(R_t\in {\mathcal R}\) and \(L_t\) is the cumulative amount of dividends paid out up to time \(t\). Given an admissible control strategy \(\pi\), the control risk process \(X_t^{\pi}\) is given by \(X_t^{\pi}=x+\int_0^t p_{R_t}d s-\sum_{i=1}^{N_t}R_{\tau_i}(U_i)-L_t\), where \(\tau_i\) is the time of occurrence of the \(i\)th claim. The corresponding ruin time of the company \(\tau^\pi=\inf\{t\geq 0:X_t^\pi<0\}\). The return function is \(V_\pi(x)=\mathbf{E}_x\left(\int_0^{\tau^\pi}\exp{(-cs)}d L_s\right)\), where \(c\) is discount factor. The optimal return function is defined as \(V(x)=\sup_{\pi\in\Pi}V_\pi(x)\).

In this paper the optimal return function \(V(x)\) is characterized as the smallest of the viscosity solutions of the associated HJB equation [see M. G. Crandall and P.-L. Lions, Trans. Am. Math. Soc. 277, 1–42 (1983; Zbl 0599.35024)]. It is proved, based on this characterisation of the optimal return function \(V(x)\), that there exists an optimal admissible strategy \(\pi^\star\) and this is a stationary strategy i.e. the decision of which reinsurance policy to choose and how much to pay out as dividend depends only on the current reserve.

The optimal control problem for restricted classes of reinsurance policies are also considered. The following families of reinsurance policies are defined: \({\mathcal R}_0\) the case of no reinsurance; \({\mathcal R}_P\) the set of proportional reinsurance policies; \({\mathcal R}_{XL}\) the case of excess-of -loss reinsurance policies and \({\mathcal R}_A\) the set of all the reinsurance policies. The optimal reinsurance policy in each of the families \({\mathcal R}_P\) [cf. H. Schmidli, Scand. Actuarial J. 2001, No. 1, 55–68 (2001; Zbl 0971.91039)], \({\mathcal R}_{XL}\), and \({\mathcal R}_{A}\) is described and it is proved that the optimal dividend payment policy can be of three possible form: the incoming premium is paid out directly as dividends, no dividend is paid, or a positive amount is paid out immediately.

In the case of no reinsurance, H. Gerber [Mitt. Verein. Schweiz. Versicherungsmath. 69, 185–228 (1969; Zbl 0193.20501)] has investigated the optimal return function [see also H. Bühlmann, Mathematical methods in risk theory. Springer Verlag, Berlin (1970; Zbl 0209.23302)]. The related problems of optimal dividend payout, when the reserve process is modelled as Brownian motion, has been solved recently by S. Asmussen and M. Taksar [Insur. Math. Econ. 20, No. 1, 1–15 (1997; Zbl 1065.91529)] in the case of no reinsurance, B. Højgaard and M. Taksar [Math. Finance 9, No. 2, 153–182 (1999; Zbl 0999.91052)] in the case of proportional reinsurance, and S. Asmussen, B. Højgaard and M. Taksar [Finance Stoch. 4, No. 3, 299–324 (2000; Zbl 0958.91026)] and T. M. Choulli, M. Taksar and X. Y. Zhou [Quant. Finance 1, 573–596 (2001)] in the case of excess-of-loss reinsurance.

Reviewer: Krzysztof Szajowski (Wrocław)

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

49L20 | Dynamic programming in optimal control and differential games |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

60G07 | General theory of stochastic processes |

93E20 | Optimal stochastic control |

### Keywords:

Cramér-Lundberg process; dividend payouts; insurance; reinsurance; Hamilton-Jacobi-Bellman equation; viscosity solution; risk control; dynamic programming principle### Citations:

Zbl 0599.35024; Zbl 0971.91039; Zbl 0193.20501; Zbl 0209.23302; Zbl 1065.91529; Zbl 0999.91052; Zbl 0958.91026
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\textit{P. Azcue} and \textit{N. Muler}, Math. Finance 15, No. 2, 261--308 (2005; Zbl 1136.91016)

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