Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles
RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 1, pp. 179-206.

This paper considers the optimal investment-reinsurance strategy in a risk model with two dependent classes of insurance business under two kinds of premium principles, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the expected value premium principle and the variance premium principle, we use the stochastic optimal control theory to derive the optimal strategy and the value function for the compound Poisson risk model as well as for the Brownian motion diffusion risk model. In particular, we find that the optimal investment strategy on the risky asset is independent to the reinsurance strategy and the reinsurance strategy for the compound Poisson risk model are very different from those for the diffusion model under both two kinds of premium principles, but the investment strategies are the same in this two risk models. Finally, numerical examples are presented to show the impact of model parameters in the optimal strategies.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2019010
Classification : 62P05, 91B30, 93E20
Mots-clés : Dependent risk, HJB equation, optimal investment-reinsurance, exponential utility, compound Poisson process
Bi, Junna 1 ; Chen, Kailing 1

1 
@article{RO_2019__53_1_179_0,
     author = {Bi, Junna and Chen, Kailing},
     title = {Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {179--206},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {1},
     year = {2019},
     doi = {10.1051/ro/2019010},
     mrnumber = {3908714},
     zbl = {1418.62373},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ro/2019010/}
}
TY  - JOUR
AU  - Bi, Junna
AU  - Chen, Kailing
TI  - Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2019
SP  - 179
EP  - 206
VL  - 53
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro/2019010/
DO  - 10.1051/ro/2019010
LA  - en
ID  - RO_2019__53_1_179_0
ER  - 
%0 Journal Article
%A Bi, Junna
%A Chen, Kailing
%T Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2019
%P 179-206
%V 53
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro/2019010/
%R 10.1051/ro/2019010
%G en
%F RO_2019__53_1_179_0
Bi, Junna; Chen, Kailing. Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 1, pp. 179-206. doi : 10.1051/ro/2019010. http://archive.numdam.org/articles/10.1051/ro/2019010/

C. Bernard, W. Tian, Optimal reinsurance arrangements under tail risk measures. J. Risk. Insur. 76 (2009) 709–725. | DOI

J. Bi, Q. Meng, Optimal investment with transaction costs and dividends for an insurer. RAIRO: OR 50 (2016) 845–855. | DOI | Numdam | MR | Zbl

S. Browne, Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20 (1995) 937–958. | DOI | MR | Zbl

J. Cai, K. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures. Astin. Bull. 37 (2007) 93–112. | DOI | MR | Zbl

J. Cai, K. Tan, C. Weng, Y. Zhang, Optimal reinsurance under VaR and CTE risk measures. Insur. Math. Econ. 5 (2008) 169–182. | MR | Zbl

M. Centeno, Measuring the effects of reinsurance by the adjustment coefficient. Insur. Math. Econ. 5 (1986) 169–182. | DOI | MR | Zbl

M. Centeno, Excess of loss reinsurance and Gerber’s inequalilty in the Sparre Anderson model. Insur. Math. Econ. 31 (2002) 415–427. | DOI | MR | Zbl

W. Fleming, H. Soner, Controlled Markov Processes and Viscosity Solutions. 2nd edition, Springer-Verlag, New York, NY (2006). | MR | Zbl

M. Hald, H. Schmidli, On the maximization of the adjustment coefficient under proportional reinsurance. Astin. Bull. 34 (2004) 75–83. | DOI | MR | Zbl

C. Hipp, M. Plum, Optimal investment for insurers. Insur. Math. Econ. 27 (2000) 215–228. | DOI | MR | Zbl

Y. Huang, Y. Ouyang, L. Tang, J. Zhou, Robust optimal investment and reinsurance problem for the product of the insurer’s and the reinsurer’s utilities. J. Comput. Appl. Math. 344 (2018) 532–552. | DOI | MR | Zbl

Y. Huang, X. Yang, J. Zhou, Robust optimal investment and reinsurance problem for a general insurance company under Heston model. Math. Method. Oper. Res. 85 (2017) 305–326. | DOI | MR | Zbl

C. Irgens, J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios. Insur. Math. Econ. 35 (2004) 21–51. | DOI | MR | Zbl

D. Li, Y. Zeng, H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps. Scand. Actuar. J. 2018 (2018) 145–171. | DOI | MR | Zbl

Z. Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion. Acta. Math. Appl. Sin-E. 23 (2007) 477–488. | DOI | MR | Zbl

Z. Liang, J. Guo, Optimal proportional reinsurance and ruin probability. Stoch. Models. 23 (2007) 333–350. | DOI | MR | Zbl

Z. Liang, J. Guo, Upper bound for ruin probabilities under optimal investment and proportional reinsurance. Appl. Stoch. Model. Bus. 24 (2008) 109–128. | DOI | MR | Zbl

Z. Liang, V. Young, Dividends and reinsurance under a penalty for ruin. Insur. Math. Econ. 50 (2012) 437–445. | DOI | Zbl

Z. Liang, K.C. Yuen, Optimal dynamic reinsurance with dependent risks: variance premium principle. Scand. Actuar. J. 2016 (2016) 18–36. | DOI | MR | Zbl

Z. Liang, K.C. Yuen, J. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process. Insur. Math. Econ. 49 (2011) 207–215. | DOI | MR | Zbl

S. Luo, M. Taksar, A. Tsoi, On reinsurance and investment for large insurance protfolios. Insur. Math. Econ. 42 (2008) 434–444. | DOI | MR | Zbl

D. Promislow, V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift. N. Am. Actuar. J. 9 (2005) 109–128. | MR | Zbl

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuar. J. 1 (2001) 55–68. | DOI | MR | Zbl

H. Schmidli, On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12 (2002) 890–907. | DOI | MR | Zbl

Z. Sun, J. Guo, Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility. Math. Method. Oper. Res. 88 (2018) 59–79. | DOI | MR | Zbl

H. Yang, L. Zhang, Optimal investment for insurer with jump-diffusion risk process. Insur. Math. Econ. 37 (2005) 615–634. | DOI | MR | Zbl

K.C. Yuen, Z. Liang, M. Zhou, Optimal proportional reinsurance with common shock dependence. Insur. Math. Econ. 64 (2015) 1–13. | DOI | MR | Zbl

Cité par Sources :