Time-consistent reinsurance-investment strategy for mean-variance insurers with defaultable security and jumps

Yıl 2018, Cilt: 47 Sayı: 3, 763 - 781, 01.06.2018

Qiang Zhang Qianqian Cui Ping Chen

Öz

This paper studies an optimal reinsurance-investment problem for a mean-variance insurer with defaultable security and jumps. Specially, we assume that the risky asset's price process is described by a geometric Lévy process. By using a game theoretic approach, we establish the extended Hamilton-Jacobi-Bellman system for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed-from expressions for the time-consistent reinsurance-investment strategy and the corresponding value function. Finally, we provide numerical examples to illustrate the impacts of model parameters on the time-consistent strategy.

Anahtar Kelimeler

Mean-variance, Proportional reinsurance, Time-consistent strategy, Defaultable bond, Geometric Lévy process

Kaynakça

• Bai, L. H., Zhang, H. Y.Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research 68 (1), 181-205, 2008.
• Bäuerle N. Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research 62 (1), 159-165, 2005.
• Bi, J. N, Meng, Q. B., Zhang, Y. J. Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer, Annals of Operations Research 212 (1), 43-59, 2014.
• Bielecki, T. R., Jang, I. Portfolio optimization with a defaultable security, Asia-Pacic Financial Markets 13 (2), 113-127, 2006.
• Björk, T., Murgoci, A. A general theory of Markovian time inconsistent stochastic control problems, Working Paper, Stockholm School of Economics, 2009.
• Björk, T., Murgoci, A., Zhou, X. Y. Mean-variance portfolio optimization with state- dependent risk aversion, Mathematical Finance 24 (1), 1-24, 2014.
• Browne, S. Optimal investment policies for a rm with random risk process: exponential utility and minimizing the probability of ruin, Mathematics of Operations Research 20, 937-958, 1995.
• Bo, L., Tang, D., Wang, Y., et al. On the conditional default probability in a regulated market: a structural approach, Quantitative Finance 11 (12), 1695-1702, 2011.
• Capponi, A., Figueroa-López, J. E. Dynamic portfolio optimization with a defaultable secu- rity and regime-switching, Mathematical Finance 24 (2),207-249, 2014.
• Chen, P., Yam, S. C. P. Optimal proportional reinsurance and investment with regime- switching for mean-variance insurers, Insurance: Mathematics and Economics 53 (3), 871- 883, 2013.
• Due, D., Singleton, K. J. Credit risk: pricing, management, and measurement, Princeton University Press, Princeton., 2003.
• Hipp, C., Plum, M. Optimal investment for insurers, Insurance: Mathematics and Economics 27, 215-228, 2000.
• Kryger, E. M., Steensen, M. Some solvable portfolio problems with quadratic and collective objectives, Working Paper, University of Copenhagen, 2010.
• Li, D. P, Rong, X. M., Zhao, H. Time-consistent reinsurance investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, Journal of Computational and Applied Mathematics 283, 142-162, 2015.
• Korn, R. Kraft, H. Optimal portfolios with defaultable securities a rm value approach, International Journal of Theoretical and Applied Finance 6 (08), 793-819, 2003.
• Promislow, D. S., Young, V. R. Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal 9, 109-128, 2005.
• Shen, Y., Zeng, Y. Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance: Mathematics and Economics 57, 1-12, 2014.
• Shen, Y., Zeng, Y. Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance: Mathematics and Economics 62, 118-137, 2015.
• Zeng, Y., Li, Z. F., Liu, J. J. Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers, Journal of Industrial and Management Optimization 6 (3), 483-496, 2010.
• Zeng, Y., Li, Z. F. Optimal time-consistent investment and reinsurance policies for mean- variance insurers, Insurance: Mathematics and Economics 49 (1),145-154, 2011.
• Zeng, Y., Li, Z. F., Lai, Y. Z. Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance: Mathematics and Economics 52 (3), 498- 507, 2013.
• Zhao, H., Shen, Y., Zeng, Y. Time-consistent investment-reinsurance strategy for mean- variance insurers with a defaultable security, Journal of Mathematical Analysis and Applications 437, 1036-1057, 2016.
• Zhu, H., Deng, C., Yue, S., et al. Optimal reinsurance and investment problem for an insurer with counterparty risk, Insurance: Mathematics and Economics 61, 242-254, 2015.