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**Optimal dividend-payout in random discrete time.**
*(English)*
Zbl 1233.91139

An insurance risk process is modelled by a Lévy process \(\{S_t\}\). It is assumed that either there is a Brownian component, there is infinite activity, or that the claim size distribution is absolutely continuous. This ensures that the distribution of the process at time \(t\) has at most one atom. The process is observed at the occurrence times \(0=Z_0 < Z_1 < \cdots\) of a renewal process. At these times, a dividend may be paid. Let \((T_n,Y_n)\) denote the inter-occurrence times and the increment of the Lévy-process between the occurrence times, respectively. The dividend payment at time \(Z_n\) is denoted by \(f(X_n) \geq 0\). The pre-dividend surplus is
\[
X_{n+1} = X_n - f(X_n) + Y_n\;,
\]
where \(X_0 = S_0\). The quantity to maximise is the expected discounted dividend payments
\[
V_f(X_0) = E\Bigl[ \sum_{k=0}^{\tau-1} e^{-\delta Z_n} f(X_n)\Bigr]\;,
\]
where \(\tau = \inf\{n: X_n < 0\}\) is the time of ruin. The value function becomes \(V(x) = \sup_f V_f(x)\).

The first part is inspired by the ideas of H. U. Gerber [Mitt. Verein. Schweiz. Versicherungsmath. 69, 185–228 (1969; Zbl 0193.20501)], see also [H. Schmidli, Stochastic control in insurance. Probability and its Applications. London: Springer (2008; Zbl 1133.93002)]. An upper and a lower bound for the value function is obtained, the Bellman equation \[ V(x) = \sup_{a \in [0,x]} \bigl\{a + E[e^{-\delta T_1} V(x-a+Y_1) ; x-a+Y_1 \geq 0]\bigr\} \] is proved, and it is observed that the optimal strategy is given by the argument, where the supremum is taken. It is further shown that the optimal strategy is a band strategy, that is, there are two disjoint sets \(A\), \(B\) with \(A \cup B = [0,\infty)\) such that \(f(x) = 0\) for \(x \in B\), and \(f(x) = \inf\{ z \in [0,x] : x-z \in B \cup\{0\}\}\). It is further verified that the set \(A\) is bounded. Thereby, the proofs are slightly more technical since the space is not discrete as in Gerber’s work.

Particular emphasis is given to the compound Poisson risk model and observation times following a Poisson process. In the case of exponentially distributed claim sizes, it is verified that the optimal strategy is of barrier type; i.e., \(A=[0,b]\) and \(B=(b,\infty)\) for some \(b \geq 0\). Further examples are given where the optimal strategy is not of barrier type. Finally, a series of compound Poisson models with exponentially distributed claim sizes is considered such that these models converge weakly to a Brownian motion with drift. It is then shown that also the value functions and the optimal strategies converge, and it is verified that the resulting barrier strategy is optimal for the diffusion case.

The first part is inspired by the ideas of H. U. Gerber [Mitt. Verein. Schweiz. Versicherungsmath. 69, 185–228 (1969; Zbl 0193.20501)], see also [H. Schmidli, Stochastic control in insurance. Probability and its Applications. London: Springer (2008; Zbl 1133.93002)]. An upper and a lower bound for the value function is obtained, the Bellman equation \[ V(x) = \sup_{a \in [0,x]} \bigl\{a + E[e^{-\delta T_1} V(x-a+Y_1) ; x-a+Y_1 \geq 0]\bigr\} \] is proved, and it is observed that the optimal strategy is given by the argument, where the supremum is taken. It is further shown that the optimal strategy is a band strategy, that is, there are two disjoint sets \(A\), \(B\) with \(A \cup B = [0,\infty)\) such that \(f(x) = 0\) for \(x \in B\), and \(f(x) = \inf\{ z \in [0,x] : x-z \in B \cup\{0\}\}\). It is further verified that the set \(A\) is bounded. Thereby, the proofs are slightly more technical since the space is not discrete as in Gerber’s work.

Particular emphasis is given to the compound Poisson risk model and observation times following a Poisson process. In the case of exponentially distributed claim sizes, it is verified that the optimal strategy is of barrier type; i.e., \(A=[0,b]\) and \(B=(b,\infty)\) for some \(b \geq 0\). Further examples are given where the optimal strategy is not of barrier type. Finally, a series of compound Poisson models with exponentially distributed claim sizes is considered such that these models converge weakly to a Brownian motion with drift. It is then shown that also the value functions and the optimal strategies converge, and it is verified that the resulting barrier strategy is optimal for the diffusion case.

Reviewer: Hanspeter Schmidli (Köln)

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

93E20 | Optimal stochastic control |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60J05 | Discrete-time Markov processes on general state spaces |

### Keywords:

stochastic control; Lévy model; discrete dividend strategy; insurance risk; Cramér–Lundberg model; Markov decision process
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\textit{H. Albrecher} et al., Stat. Risk. Model. 28, No. 3, 251--276 (2011; Zbl 1233.91139)

### References:

[1] | Albrecher H., RACSAM Rev. R. Acad. Cien. Serie A. Mat. 103 (2) pp 295– (2009) · Zbl 1187.93138 |

[2] | Azcue P., Math. Finance 15 (2) pp 261– (2005) · Zbl 1136.91016 |

[3] | Bäuerle N., Math. Finance 14 (1) pp 99– (2004) · Zbl 1097.91052 |

[4] | Gerber H. U., Schweiz. Aktuarver. Mitt. 69 pp 185– (1969) |

[5] | Loeffen R., Insurance Math. Econom. 46 (1) pp 98– (2010) · Zbl 1231.91212 |

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