##
**Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints.**
*(English)*
Zbl 1156.91391

Summary: We consider that the surplus of an insurance company follows a Cramér-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound \(a\). For instance the case \(a=1\) means that the management cannot borrow money to buy stocks.

C. Hipp and M. Plum [Insur. Math. Econ. 28, No. 2, 215–228 (2000; Zbl 1007.91025)] and [H. Schmidli, Ann. Appl. Probab. 12, No. 3, 890–907 (2002; Zbl 1021.60061)] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.

We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.

C. Hipp and M. Plum [Insur. Math. Econ. 28, No. 2, 215–228 (2000; Zbl 1007.91025)] and [H. Schmidli, Ann. Appl. Probab. 12, No. 3, 890–907 (2002; Zbl 1021.60061)] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.

We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91B28 | Finance etc. (MSC2000) |

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

### Keywords:

Cramér-Lundberg process; ruin probability; insurance; portfolio optimization; borrowing constraints; Hamilton-Jacobi-Bellman equation
PDF
BibTeX
XML
Cite

\textit{P. Azcue} and \textit{N. Muler}, Insur. Math. Econ. 44, No. 1, 26--34 (2009; Zbl 1156.91391)

### References:

[1] | Azcue, P., Muler, N., 2008. Optimal portfolio and dividend distribution policies in an insurance company (submitted for publication) · Zbl 1196.91033 |

[2] | Bayraktar, E.; Young, V.R., Minimizing the probability of lifetime ruin under borrowing constraints, Insurance: mathematics and economics, 41, 196-221, (2007) · Zbl 1119.91041 |

[3] | 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 |

[4] | Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, (1993), Springer-Verlag New York · Zbl 0773.60070 |

[5] | Frovola, A.G.; Kabanov, Yu.M.; Pergamenshchikov, S.M., In the insurance business risky investments are dangerous, Finance and stochastics, 6, 227-235, (2002) · Zbl 1002.91037 |

[6] | Gaier, J.; Grandits, P., Ruin probabilities in the presence of regularly varying tails and optimal investment, Insurance: mathematics and economics, 30, 211-217, (2002) · Zbl 1055.91049 |

[7] | Hipp, C.; Plum, M., Optimal investment for insurers, Insurance: mathematics and economics, 27, 215-228, (2000) · Zbl 1007.91025 |

[8] | Luo, S., Ruin minimization for insurers with borrowing constraints, North American actuarial journal, 12, 143-174, (2008) |

[9] | Paulsen, J.; Gjessing, H.K., Ruin theory with stochastic return on investments, Advances in applied probability, 29, 965-985, (1997) · Zbl 0892.90046 |

[10] | Paulsen, J., Sharp conditions for certain ruin in a risk process with stochastic return on investments, Stochastic processes and their applications, 75, 135-148, (1998) · Zbl 0932.60044 |

[11] | Promislow, D.; Young, V.R., Minimizing the probability of ruin when claims follow Brownian motion with drift, North American actuarial journal, 9, 3, 109-128, (2005) · Zbl 1141.91543 |

[12] | Protter, P., Stochastic integration and differential equations, (1992), Springer-Verlag Berlin |

[13] | Schmidli, H., On minimizing the ruin probability by investment and reinsurance, Ann. appl. probab., 12, 890-907, (2002) · Zbl 1021.60061 |

[14] | Schmidli, H., Stochastic control in insurance, (2008), Springer-Verlag London · Zbl 1133.93002 |

[15] | Vila, J.; Zariphopoulou, T., Optimal consumption and portfolio choice with borrowing constraints, Journal of economic theory, 77, 402-431, (1997) · Zbl 0897.90078 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.