## Optimal dividend strategies in a Cramér-Lundberg model with capital injections and administration costs.(English)Zbl 1222.91026

The authors consider a classical risk model with dividend payments and capital injections in the presence of both fixed and proportional administration costs. In this model, a dividends-injections strategy is admissible if it does not lead to a negative surplus or ruin. The value of a strategy is the discounted value of the dividends minus the costs. The authors argue that it can not be optimal to make a capital injections unless the surplus is negative. Also, at the time of an injection the company may not only inject the deficit, but inject additional capital $$C\geq 0$$ to prevent future capital injections. The associated Hamilton-Jacobi-Bellman equation is derived and it is shown that the optimal strategy is of band type. By using Gerber-Shiu functions, the authors derive a method to determine numerically the solution to the integro-differential equation and the unknown value $$C$$.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60H30 Applications of stochastic analysis (to PDEs, etc.)
Full Text:

### References:

 [1] Albrecher H, Thonhauser S (2008) Optimal dividend strategies for a risk process under force of interest. Insur Math Econ 43(1):134–149 · Zbl 1140.91371 [2] Albrecher H, Thonhauser S (2009) Optimality results for dividend problems in insurance. RACSAM Rev R Acad Cien Serie A Math 103(2):295–320 · Zbl 1187.93138 [3] Asmussen S, Taksar M (1997) Controlled diffusion models for optimal dividend pay-out. Insur Math Econ 20(1):1–15 · Zbl 1065.91529 [4] Avanzi B (2009) Strategies for dividend distribution: a review. North Am Actuar J 13(2):217–251 [5] Avram F, Palmowski Z, Pistorius MR (2007) On the optimal dividend problem for a spectrally negative Lévy process. Ann Appl Prob 17(1):156–180 · Zbl 1136.60032 [6] Azcue P, Muler N (2005) Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15(2):261–308 · Zbl 1136.91016 [7] Brémaud P (1981) Point processes and queues. Springer, New York [8] Dickson DCM, Waters HR (2004) Some optimal dividends problems. ASTIN Bull 34(1):49–74 · Zbl 1097.91040 [9] de Finetti B (1957) Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth international congress of actuaries, vol 2, pp 433–443 [10] Evans LC (1998) Partial differential equations. AMS, Providence · Zbl 0898.35001 [11] Gerber HU (1969) Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Schweiz Verein Versicherungsmath Mitt 69:185–228 · Zbl 0193.20501 [12] Gerber HU, Lin XS, Yang H (2006) A note on the dividends-penalty identity and the optimal dividend barrier. ASTIN Bull 36:489–503 · Zbl 1162.91374 [13] Gerber HU, Shiu ESW (1998) On the time value of ruin. North Am Actuar J 8(1):1–20 · Zbl 1085.62122 [14] Gerber HU, Shiu ESW, Smith N (2006) Maximizing dividends without bankruptcy. ASTIN Bull 36:5–23 · Zbl 1162.91375 [15] He L, Liang Z (2009) Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs. Insur Math Econ 44(1):88–94 · Zbl 1156.91395 [16] Jeanblanc-Piqué M, Shiryaev AN (1995) Optimization of the flow of dividends. Uspekhi Mat Nauk 50(2(302)):25–46 · Zbl 0878.90014 [17] Kulenko N, Schmidli H (2008) Optimal dividend strategies in a Cramér–Lundberg model with capital injections. Insur Math Econ 43(2):270–278 · Zbl 1189.91075 [18] Lokka A, Zervos M (2008) Optimal dividend and insurance of equity policies in the presence of proportional costs. Insur Math Econ 42(3):954–961 · Zbl 1141.91528 [19] Paulsen J (2008) Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs. SIAM J Control Optim 47(5):2201–2226 · Zbl 1171.49027 [20] Schmidli H (2008) Stochastic control in insurance. Springer, London · Zbl 1133.93002 [21] Shreve SE, Lehoczky JP, Gaver DP (1984) Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J Control Optim 22:55–75 · Zbl 0535.93071 [22] Thonhauser S, Albrecher H (2007) Dividend maximization under consideration of the time value of ruin. Insur Math Econ 41(1):163–184 · Zbl 1119.91047
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.