Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

OPTIMAL INVESTMENT FOR THE INSURER IN THE LEVY MARKET UNDER THE MEAN-VARIANCE CRITERION

  • Liu, Junfeng (Department of Mathematics, East China University of Science and Technology)
  • Received : 2009.09.10
  • Accepted : 2009.10.25
  • Published : 2010.05.30
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Abstract

In this paper we apply the martingale approach, which has been widely used in mathematical finance, to investigate the optimal investment problem for an insurer under the criterion of mean-variance. When the risk and security assets are described by the L$\acute{e}$vy processes, the closed form solutions to the maximization problem are obtained. The mean-variance efficient strategies and frontier are also given.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20(1995), 937-958. https://doi.org/10.1287/moor.20.4.937
  2. C. Hipp and M. Plum, Optimal investment for insurers. Insur. Math. Econ. 27(2000), 215-228. https://doi.org/10.1016/S0167-6687(00)00049-4
  3. C. Hipp, Stochastic control with application in insurance. In: Stochastic Methods in Finance. In: Lecture Notes in Mathematics, vol. 1856(2004). Springer, Berlin, pp. 127-164.
  4. C.S. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin. North American Actuarial Journal 8(2004), 11-31. https://doi.org/10.1080/10920277.2004.10596134
  5. H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process. Insur. Math. Econ. 37(2005), 615-634.I https://doi.org/10.1016/j.insmatheco.2005.06.009
  6. N. Wang, Optimal investment for an insurer with exponential utility preference. Insur. Math. Econ. 40(2007), 77-84. https://doi.org/10.1016/j.insmatheco.2006.02.008
  7. Z. Wang, J. Xia and L. Zhang, Optimal investment for an insurer: The martingale approach. Insur. Math. Econ. 40(2007), 322-334. https://doi.org/10.1016/j.insmatheco.2006.05.003
  8. H. Markowitz, Portfolio selection. J. Finance 7(1952), 77-91. https://doi.org/10.2307/2975974
  9. R.C. Merton, An analytic derivation of the efficient frontier. J. Finance Quant. Anl. 7(1972), 1851-1872. https://doi.org/10.2307/2329621
  10. D. Li and W.L. Ng, Optimal dynamic portfolio selection: multiperiod mean-variance formulation. Math. Finance 10(2000), 387-406. https://doi.org/10.1111/1467-9965.00100
  11. D. Kramkov, W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(1999), 904-950. https://doi.org/10.1214/aoap/1029962818
  12. R. Cont and P. Tankov, Financial Modelling With Jump Processes. In: Chapman and Hall/CRC Financial Mathematics Series, 2003.
  13. X. Zhou and D. Li, Continuous-time mean-variance portfolio selection: a stochastic LQ framework. Appl. Math. Optim. 42(2000). 19-33. https://doi.org/10.1007/s002450010003
  14. X. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: a continuous time model. SIAM J. Control Optim. 42, 1466-1482.
  15. I. Karatzas, J. P. Lehoczky, S.E. Shreve and G. L. Xu, Martingale and duality methods for utility maximization in incomplete markets. SIAM J. Control Optim. 29(1991), 702-730. https://doi.org/10.1137/0329039
  16. X. Li, X. Zhou and A.E.B. Lim, Dynamic mean-variance portfolio selection with noshorting constraints. SIAM J. Control Optim. 40(2002), 1540-1555. https://doi.org/10.1137/S0363012900378504
  17. Q. Zhou, Optimal investment for an insurer in the Levy market: The martingale approach. Statist. Probab. lett. 79(2009), 1602-1607. https://doi.org/10.1016/j.spl.2009.03.027