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doi: 10.3934/jimo.2021015
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Time-consistent investment-reinsurance strategy with a defaultable security under ambiguous environment

Shan Liu , Hui Zhao and  Ximin Rong ,

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Ximin Rong

Received  May 2020 Revised  November 2020 Early access December 2020

Fund Project: This research was supported by National Natural Science Foundation of China (Grant Nos. 11771329, 11871052, 11971301), and Nature Science foundation of Tianjin city (20JCYBJC01160)

This paper considers an investment and reinsurance problem with a defaultable security for an insurer in an environment with parameter uncertainties. Suppose that the insurer is ambiguous about the insurance claims. Specifically, the insurance claim is exponentially distributed and the rate parameter is uncertain. The insurer is allowed to invest in a financial market consisting of a risk-free bond, a stock whose price process satisfies the Heston's SV model and a defaultable bond. Moreover, the insurer is allowed to purchase proportional reinsurance and aims to maximize the smooth ambiguity utility proposed in Klibanoff et al. [15]. By applying stochastic control approach, we establish the extended HJB system and derive the time-consistent investment-reinsurance strategy for the post-default case and the pre-default case, respectively. Finally, a sensitivity analysis is provided to illustrate the effects of model parameters on the equilibrium reinsurance-investment strategy under the smooth ambiguity.

Keywords: Smooth ambiguity utility, Heston's SV model, defaultable bond, time-consistent strategy, reinsurance and investment.
Mathematics Subject Classification: Primary: 93E20; Secondary: 91G80, 91G10.
Citation: Shan Liu, Hui Zhao, Ximin Rong. Time-consistent investment-reinsurance strategy with a defaultable security under ambiguous environment. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021015
References:
[1]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[2]

T. R. Bielecki and I. Jang, Portfolio optimization with a defaultable security, Asia-Pacific Financial Markets, 13 (2007), 113-127.   Google Scholar

[3]

T. R. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer-Verlag, Berlin, Heidelberg, 2002.  Google Scholar

[4]

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.  doi: 10.1287/moor.20.4.937.  Google Scholar

[5]

Y. Cao and N. Wan, Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation, Insurance Math. Econom., 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006.  Google Scholar

[6]

H. Chen, M. Sherris, T. Sun and W. Zhu, Living with ambiguity: Pricing mortality-linked securities with smooth ambiguity preferences, Journal of Risk and Insurance, 80 (2013), 705-732.  doi: 10.2139/ssrn.2007342.  Google Scholar

[7]

J. C. Cox and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166.  doi: 10.1016/0304-405X(76)90023-4.  Google Scholar

[8]

C. Deng, X. Zheng and H. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, European J. Oper. Res., 264 (2018), 1144-1158.  doi: 10.1016/j.ejor.2017.06.065.  Google Scholar

[9]

A. Gu, F. G. Viens and H. Yao, Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing, Insurance Math. Econom., 80 (2018), 93-109.  doi: 10.1016/j.insmatheco.2018.03.004.  Google Scholar

[10]

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.  Google Scholar

[11]

G. Guan, Z. Liang and J. Feng, Time-consistent proportional reinsurance and investment strategies under ambiguous environment, Insurance Math. Econom., 83 (2018), 122-133.  doi: 10.1016/j.insmatheco.2018.09.007.  Google Scholar

[12]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[13]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance Math. Econom., 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[14]

D. G. Hobson and L. C. G. Rogers, Complete models with stochastic volatility, Math. Finance, 8 (1998), 27-48.  doi: 10.1111/1467-9965.00043.  Google Scholar

[15]

P. Klibanoff, M. Marinacci and S. Mukerji, A smooth model of decision making under ambiguity, Econometrica, 73 (2005), 1849-1892.  doi: 10.1111/j.1468-0262.2005.00640.x.  Google Scholar

[16]

Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.  Google Scholar

[17]

Z. Liang, K. C. Yuen and K. C. Cheung, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Appl. Stoch. Models Bus. Ind., 28 (2012), 585-597.  doi: 10.1002/asmb.934.  Google Scholar

[18]

X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV model, N. Am. Actuar. J., 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.  Google Scholar

[19]

C. S. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, N. Am. Actuar. J., 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar

[20]

S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar

[21]

Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, J. Math. Anal. Appl., 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.  Google Scholar

[22]

Z. Wang, J. Xia and L. Zhang, Optimal investment for an insurer: The martingale approach, Insurance Math. Econom., 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003.  Google Scholar

[23]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[24]

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), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011.  Google Scholar

[25]

Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001.  Google Scholar

[26]

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.  Google Scholar

[27]

H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.  Google Scholar

[28]

H. Zhao, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, J. Math. Anal. Appl., 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035.  Google Scholar

[29]

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.  doi: 10.1016/j.insmatheco.2015.12.008.  Google Scholar

[30]

H. Zhu, C. Deng, S. Yue and Y. Deng, Optimal reinsurance and investment problem for an insurer with counterparty risk, Insurance Math. Econom., 61 (2015), 242-254.  doi: 10.1016/j.insmatheco.2015.01.013.  Google Scholar

show all references

References:
[1]

L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[2]

T. R. Bielecki and I. Jang, Portfolio optimization with a defaultable security, Asia-Pacific Financial Markets, 13 (2007), 113-127.   Google Scholar

[3]

T. R. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer-Verlag, Berlin, Heidelberg, 2002.  Google Scholar

[4]

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.  doi: 10.1287/moor.20.4.937.  Google Scholar

[5]

Y. Cao and N. Wan, Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation, Insurance Math. Econom., 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006.  Google Scholar

[6]

H. Chen, M. Sherris, T. Sun and W. Zhu, Living with ambiguity: Pricing mortality-linked securities with smooth ambiguity preferences, Journal of Risk and Insurance, 80 (2013), 705-732.  doi: 10.2139/ssrn.2007342.  Google Scholar

[7]

J. C. Cox and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166.  doi: 10.1016/0304-405X(76)90023-4.  Google Scholar

[8]

C. Deng, X. Zheng and H. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, European J. Oper. Res., 264 (2018), 1144-1158.  doi: 10.1016/j.ejor.2017.06.065.  Google Scholar

[9]

A. Gu, F. G. Viens and H. Yao, Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing, Insurance Math. Econom., 80 (2018), 93-109.  doi: 10.1016/j.insmatheco.2018.03.004.  Google Scholar

[10]

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.  Google Scholar

[11]

G. Guan, Z. Liang and J. Feng, Time-consistent proportional reinsurance and investment strategies under ambiguous environment, Insurance Math. Econom., 83 (2018), 122-133.  doi: 10.1016/j.insmatheco.2018.09.007.  Google Scholar

[12]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[13]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance Math. Econom., 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[14]

D. G. Hobson and L. C. G. Rogers, Complete models with stochastic volatility, Math. Finance, 8 (1998), 27-48.  doi: 10.1111/1467-9965.00043.  Google Scholar

[15]

P. Klibanoff, M. Marinacci and S. Mukerji, A smooth model of decision making under ambiguity, Econometrica, 73 (2005), 1849-1892.  doi: 10.1111/j.1468-0262.2005.00640.x.  Google Scholar

[16]

Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.  Google Scholar

[17]

Z. Liang, K. C. Yuen and K. C. Cheung, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Appl. Stoch. Models Bus. Ind., 28 (2012), 585-597.  doi: 10.1002/asmb.934.  Google Scholar

[18]

X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV model, N. Am. Actuar. J., 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.  Google Scholar

[19]

C. S. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, N. Am. Actuar. J., 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar

[20]

S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar

[21]

Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, J. Math. Anal. Appl., 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.  Google Scholar

[22]

Z. Wang, J. Xia and L. Zhang, Optimal investment for an insurer: The martingale approach, Insurance Math. Econom., 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003.  Google Scholar

[23]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[24]

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), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011.  Google Scholar

[25]

Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001.  Google Scholar

[26]

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.  Google Scholar

[27]

H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.  Google Scholar

[28]

H. Zhao, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, J. Math. Anal. Appl., 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035.  Google Scholar

[29]

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.  doi: 10.1016/j.insmatheco.2015.12.008.  Google Scholar

[30]

H. Zhu, C. Deng, S. Yue and Y. Deng, Optimal reinsurance and investment problem for an insurer with counterparty risk, Insurance Math. Econom., 61 (2015), 242-254.  doi: 10.1016/j.insmatheco.2015.01.013.  Google Scholar

Figure 1.  Effects of $ k $ and $ \alpha $ on $ a^{*} $
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Figure 2.  Effects of $ \nu_1 $ and $ q^{\nu}_{1} $ on $ a^{*} $
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Figure 3.  Effects of $ \lambda $ and $ \theta $ on $ a^{*} $
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Figure 4.  Effects of $ k $ and $ \xi $ on $ \pi_{s}^{*} $
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Figure 5.  The Effect of the $ \rho $ on $ \pi_{s}^{*} $
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Figure 6.  The Effect of parameter $ \gamma $ on $ \pi_{s}^{*} $
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Figure 7.  The Effect of parameter $ \sigma $ on $ \pi_{s}^{*} $
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Figure 8.  Effects of parameters $ \alpha, k, h^{\mathrm{P}}, \zeta $ and $ \delta $ on $ \pi^{*}_{p} $
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