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**Optimal investment for insurers.**
*(English)*
Zbl 1007.91025

Insurance business considered is modelled by a compound Poisson process with the Black-Scholes type market index. The authors show that the ruin probability of this risk process is minimized by the choice of a suitable investment strategy for a capital market index. Let \(T(t)\), \(t \geq 0,\) be the surplus process. The optimal invested amount \(A_t\), \(t \geq 0,\) at time \(t\) has the following properties: the amount of money \(A_t = A(T(t))\); \(A(0) = 0\); the derivative \(A'\) has a pole at \(0\); the function \(A\) remains bounded for exponential claim sizes, and it is unbounded for heavy-tailed claim size distributions. The result is obtained with the aid of the Bellman equation - a second order nonlinear integro-differential equation - which characterizes the value function and the optimal strategy. More explicit solutions are determined when the claim size distribution is exponential, in which case a numerical example is also provided. Another example refers to the case of Pareto claim size.

Using in the model a Brownian motion with drift in place of the compound Poisson process, S. Browne [Meth. Oper. Res. 20, 937-958 (1995; Zbl 0846.90012)] obtained the quite different result: the optimal strategy is the investment of a constant amount of money in the risky asset, irrespectively of the size of the surplus.

Using in the model a Brownian motion with drift in place of the compound Poisson process, S. Browne [Meth. Oper. Res. 20, 937-958 (1995; Zbl 0846.90012)] obtained the quite different result: the optimal strategy is the investment of a constant amount of money in the risky asset, irrespectively of the size of the surplus.

Reviewer: Bogdan Choczewski (Kraków)

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

93E20 | Optimal stochastic control |

60G40 | Stopping times; optimal stopping problems; gambling theory |

### Keywords:

stochastic control theory; compound Poisson process; geometric Brownian motion; Bellman’s equation; investment; ruin probability### Citations:

Zbl 0846.90012
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\textit{C. Hipp} and \textit{M. Plum}, Insur. Math. Econ. 27, No. 2, 215--228 (2000; Zbl 1007.91025)

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### References:

[1] | Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics of operations research, 20, 937-958, (1995) · Zbl 0846.90012 |

[2] | Fleming, W.H., Soner, M., 1993. Controlled Markov Processes and Viscosity Solutions. Springer, New York. · Zbl 0773.60070 |

[3] | Hipp, C., Taksar, M., 2000. Stochastic control for optimal new business. Insurance: Mathematics and Economics 26, 185-192. · Zbl 1103.91366 |

[4] | Hoejgaard, B., Taksar, M., 1998. Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal, 166-180. · Zbl 1075.91559 |

[5] | Schmidli, H., 1999. Optimal proportional reinsurance policies in a dynamic setting. Research Report 403. Department of Theoretical Statistics, Aarhus University, Denmark. · Zbl 0971.91039 |

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