# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019011

## Optimal reinsurance-investment problem with dependent risks based on Legendre transform

 1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 2 College of Science, Army Engineering University of PLA, Nanjing 211101, China

* Corresponding author

Received  April 2018 Revised  October 2018 Published  March 2019

Fund Project: This work was supported by NNSF of China (No.11871275; No.11371194)

This paper investigates an optimal reinsurance-investment problem in relation to thinning dependent risks. The insurer's wealth process is described by a risk model with two dependent classes of insurance business. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price follows CEV model. Our aim is to maximize the expected exponential utility of terminal wealth. Applying Legendre transform-dual technique along with stochastic control theory, we obtain the closed-form expression of optimal strategy. In addition, our wealth process will reduce to the classical Cramér-Lundberg (C-L) model when $p = 0$, in this case, we achieve the explicit expression of the optimal strategy for Hyperbolic Absolute Risk Aversion (HARA) utility by using Legendre transform. Finally, some numerical examples are presented to illustrate the impact of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategy.

Citation: Yan Zhang, Peibiao Zhao. Optimal reinsurance-investment problem with dependent risks based on Legendre transform. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019011
##### References:
 [1] L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205. doi: 10.1007/s00186-007-0195-4. [2] J. Bi, Z. Liang and F. Xu, Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance Math. Econom., 70 (2016), 245-258. doi: 10.1016/j.insmatheco.2016.06.012. [3] H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227. doi: 10.1016/j.insmatheco.2016.10.014. [4] J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18. doi: 10.1016/j.insmatheco.2009.02.006. [5] L. Gong, A. Badescu and E. Cheung, Recursive methods for a multidimensional risk process with common shocks, Insurance Math. Econom., 50 (2012), 109-120. doi: 10.1016/j.insmatheco.2011.10.007. [6] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9. [7] M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance Math. Econom., 33 (2003), 611-627. doi: 10.1016/j.insmatheco.2003.09.003. [8] M. Gu, Y. Yang, S. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587. doi: 10.1016/j.insmatheco.2010.03.001. [9] E. Jung and J. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673. doi: 10.1016/j.insmatheco.2012.09.009. [10] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162. doi: 10.1016/j.cam.2015.01.038. [11] Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 1 (2016), 18-36. doi: 10.1080/03461238.2014.892899. [12] Z. Liang, J. Bi, K. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Math. Methods Oper. Res., 84 (2016), 155-181. doi: 10.1007/s00186-016-0538-0. [13] X. Liang and G. Wang, On a reduced form credit risk model with common shock and regime switching, Insurance Math. Econom., 51 (2012), 567-575. doi: 10.1016/j.insmatheco.2012.07.010. [14] Z. Liang and M. Long, Minimization of absolute ruin probability under negative correlation assumption, Insurance Math. Econom., 65 (2015), 247-258. doi: 10.1016/j.insmatheco.2015.10.003. [15] J. Liu, Portfolio selection in stochastic environments, Rev. Financ. Stud., 20 (2007), 1-39. doi: 10.1093/rfs/hhl001. [16] S. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 110-128. doi: 10.1080/10920277.2005.10596214. [17] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance Math. Econom., 57 (2014), 1-12. doi: 10.1016/j.insmatheco.2014.04.004. [18] D. Sheng, Explicit solution of reinsurance-investment problem for an insurer with dynamic income under vasicek model, Advances in Mathematical Physics, 2016 (2016), Art. ID 1967872, 13 pp. doi: 10.1155/2016/1967872. [19] G. Wang and K. Yuen, On a correlated aggregate claims model with thinning-dependence structure, Insurance Math. Econom., 36 (2005), 456-468. doi: 10.1016/j.insmatheco.2005.04.004. [20] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 338 (2018), 414-431. doi: 10.1016/j.cam.2017.08.001. [21] J. Xiao, H. Zhai and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310. doi: 10.1016/j.insmatheco.2006.04.007. [22] K. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance Math. Econom., 64 (2015), 1-13. doi: 10.1016/j.insmatheco.2015.04.009. [23] Y. Zeng, D. Li and A. Gu, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance Math. Econom., 66 (2016), 138-152. doi: 10.1016/j.insmatheco.2015.10.012. [24] H. Zhao, X. Rong and Y. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, Insurance Math. Econom., 53 (2013), 504-514. doi: 10.1016/j.insmatheco.2013.08.004. [25] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67. doi: 10.1016/j.insmatheco.2014.06.006.

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##### References:
 [1] L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205. doi: 10.1007/s00186-007-0195-4. [2] J. Bi, Z. Liang and F. Xu, Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance Math. Econom., 70 (2016), 245-258. doi: 10.1016/j.insmatheco.2016.06.012. [3] H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227. doi: 10.1016/j.insmatheco.2016.10.014. [4] J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18. doi: 10.1016/j.insmatheco.2009.02.006. [5] L. Gong, A. Badescu and E. Cheung, Recursive methods for a multidimensional risk process with common shocks, Insurance Math. Econom., 50 (2012), 109-120. doi: 10.1016/j.insmatheco.2011.10.007. [6] J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9. [7] M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance Math. Econom., 33 (2003), 611-627. doi: 10.1016/j.insmatheco.2003.09.003. [8] M. Gu, Y. Yang, S. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587. doi: 10.1016/j.insmatheco.2010.03.001. [9] E. Jung and J. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673. doi: 10.1016/j.insmatheco.2012.09.009. [10] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162. doi: 10.1016/j.cam.2015.01.038. [11] Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 1 (2016), 18-36. doi: 10.1080/03461238.2014.892899. [12] Z. Liang, J. Bi, K. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Math. Methods Oper. Res., 84 (2016), 155-181. doi: 10.1007/s00186-016-0538-0. [13] X. Liang and G. Wang, On a reduced form credit risk model with common shock and regime switching, Insurance Math. Econom., 51 (2012), 567-575. doi: 10.1016/j.insmatheco.2012.07.010. [14] Z. Liang and M. Long, Minimization of absolute ruin probability under negative correlation assumption, Insurance Math. Econom., 65 (2015), 247-258. doi: 10.1016/j.insmatheco.2015.10.003. [15] J. Liu, Portfolio selection in stochastic environments, Rev. Financ. Stud., 20 (2007), 1-39. doi: 10.1093/rfs/hhl001. [16] S. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 110-128. doi: 10.1080/10920277.2005.10596214. [17] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance Math. Econom., 57 (2014), 1-12. doi: 10.1016/j.insmatheco.2014.04.004. [18] D. Sheng, Explicit solution of reinsurance-investment problem for an insurer with dynamic income under vasicek model, Advances in Mathematical Physics, 2016 (2016), Art. ID 1967872, 13 pp. doi: 10.1155/2016/1967872. [19] G. Wang and K. Yuen, On a correlated aggregate claims model with thinning-dependence structure, Insurance Math. Econom., 36 (2005), 456-468. doi: 10.1016/j.insmatheco.2005.04.004. [20] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 338 (2018), 414-431. doi: 10.1016/j.cam.2017.08.001. [21] J. Xiao, H. Zhai and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310. doi: 10.1016/j.insmatheco.2006.04.007. [22] K. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance Math. Econom., 64 (2015), 1-13. doi: 10.1016/j.insmatheco.2015.04.009. [23] Y. Zeng, D. Li and A. Gu, Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance Math. Econom., 66 (2016), 138-152. doi: 10.1016/j.insmatheco.2015.10.012. [24] H. Zhao, X. Rong and Y. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, Insurance Math. Econom., 53 (2013), 504-514. doi: 10.1016/j.insmatheco.2013.08.004. [25] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67. doi: 10.1016/j.insmatheco.2014.06.006.
Effect of $t$ on the optimal reinsurance strategies
Effect of $v$ on the optimal reinsurance strategies
Effect of $p$ on the optimal reinsurance strategies
Effect of $\alpha _1$ on the optimal reinsurance strategies
Effect of $\alpha _2$ on the optimal reinsurance strategies
Effect of $s$ on the optimal investment strategy
Effect of $\mu - r$ on the optimal investment strategy
Effect of $\sigma$ on the optimal investment strategy
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