Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ROBUST OPTIMAL PROPORTIONAL REINSURANCE AND INVESTMENT STRATEGY FOR AN INSURER WITH ORNSTEIN-UHLENBECK PROCESS

  • Ma, Jianjing (Center for Financial Engineering Soochow University) ;
  • Wang, Guojing (Center for Financial Engineering Soochow University) ;
  • Xing, Yongsheng (School of Mathematics and Information Science Shandong Technology and Business University)
  • Received : 2018.11.14
  • Accepted : 2019.08.23
  • Published : 2019.11.30
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Abstract

This paper analyzes a robust optimal reinsurance and investment strategy for an Ambiguity-Averse Insurer (AAI), who worries about model misspecification and insists on seeking robust optimal strategies. The AAI's surplus process is assumed to follow a jump-diffusion model, and he is allowed to purchase proportional reinsurance or acquire new business, meanwhile invest his surplus in a risk-free asset and a risky-asset, whose price is described by an Ornstein-Uhlenbeck process. Under the criterion for maximizing the expected exponential utility of terminal wealth, robust optimal strategy and value function are derived by applying the stochastic dynamic programming approach. Serval numerical examples are given to illustrate the impact of model parameters on the robust optimal strategies and the loss utility function from ignoring the model uncertainty.

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References

  1. E. Anderson, L. P. Hansen, and T. J. Sargent, Robustness detection and the price of risk, Working Paper, University of Chicago, 1988. Available at: https://files.nyu.edu/ts43/public/research/.svn/text-base/ahs3.pdf.svn-base.
  2. E. Anderson, L. P. Hansen, and T. J. Sargent, A quartet of semi-groups for model specication, robustness, prices of risk, and model detection, JEEA 1 (2003), 68-123.
  3. L. Bai and J. Guo, Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach, Front. Math. China 2 (2007), no. 4, 527-537. https://doi.org/10.1007/s11464-007-0032-3
  4. L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom. 42 (2008), no. 3, 968-975. https://doi.org/10.1016/j.insmatheco.2007.11.002
  5. N. Bauerle, Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res. 62 (2005), no. 1, 159-165. https://doi.org/10.1007/s00186-005-0446-1
  6. N. Branger and L. S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, J. Bank. Finance 37 (2013), 5036-5047. https://doi.org/10.1016/j.jbankfin.2013.08.023
  7. 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), no. 4, 937-958. https://doi.org/10.1287/moor.20.4.937
  8. Z. Liang, K. C. Yuen, and J. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance Math. Econom. 49 (2011), no. 2, 207-215. https://doi.org/10.1016/j.insmatheco.2011.04.005
  9. H. N. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Ann. Finance 6 (2010), 435-454. https://doi.org/10.1007/s10436-010-0164-4
  10. P. J. Maenhout, Robust portfolio rules and asset pricing, Rev. Financ. Stud. 17 (2004), 951-983. https://doi.org/10.1093/rfs/hhh003
  11. P. J. Maenhout, Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium, J. Econom. Theory 128 (2006), no. 1, 136-163. https://doi.org/10.1016/j.jet.2005.12.012
  12. S. Mataramvura and B. ksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics 80 (2008), no. 4, 317-337. https://doi.org/10.1080/17442500701655408
  13. R. Rishel, Optimal portfolio management with partial observations and power utility function, in Stochastic analysis, control, optimization and applications, 605-619, Systems Control Found. Appl, Birkhauser Boston, Boston, MA, 1999.
  14. H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab. 12 (2002), no. 3, 890-907. https://doi.org/10.1214/aoap/1031863173
  15. R. Uppal and T. Wang, Model misspecication and underdiversification, J. Finance 58 (2003), 2465-2486. https://doi.org/10.1046/j.1540-6261.2003.00612.x
  16. H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom. 37 (2005), no. 3, 615-634. https://doi.org/10.1016/j.insmatheco.2005.06.009
  17. B. Yi, Z. Li, F. G. Viens, and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insurance Math. Econom. 53 (2013), no. 3, 601-614. https://doi.org/10.1016/j.insmatheco.2013.08.011
  18. B. Yi, Z. Li, F. G. Viens, and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scand. Actuar. J. 2015, no. 8, 725-751. https://doi.org/10.1080/03461238.2014.883085
  19. Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom. 49 (2011), no. 1, 145-154. https://doi.org/10.1016/j.insmatheco.2011.01.001
  20. Y. Zeng, Z. Li, and Y. Lai, Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance Math. Econom. 52 (2013), no. 3, 498-507. https://doi.org/10.1016/j.insmatheco.2013.02.007
  21. X. Zheng, J. Zhou, and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model, Insurance Math. Econom. 67 (2016), 77-87. https://doi.org/10.1016/j.insmatheco.2015.12.008