# American Institute of Mathematical Sciences

September  2020, 16(5): 2563-2579. doi: 10.3934/jimo.2019070

## Optimal investment and risk control problems with delay for an insurer in defaultable market

 1 School of Finance, Guangdong University of Foreign Studies, 510006, Guangzhou, China 2 Institute of Big Data and Internet Innovation, Hunan University of Commerce, 410205, Changsha, China 3 Business School, Central South University, 410012, Changsha, China

* Corresponding author: Yan Chen

Received  July 2017 Revised  March 2019 Published  September 2020 Early access  July 2019

This paper addresses a investment and risk control problem with a delay for an insurer in the defaultable market. Suppose that an insurer can invest in a risk-free bank account, a risky stock and a defaultable bond. Taking into account the history of the insurer's wealth performance, the controlled wealth process is governed by a stochastic delay differential equation. The insurer's goal is to maximize the expected exponential utility of the combination of terminal wealth and average performance wealth. We decompose the original optimization problem into two subproblems: a pre-default case and a post-default case. The explicit solutions in a finite dimensional space are derived for a illustrative situation, and numerical illustrations and sensitivity analysis for our results are provided.

Citation: Chao Deng, Haixiang Yao, Yan Chen. Optimal investment and risk control problems with delay for an insurer in defaultable market. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2563-2579. doi: 10.3934/jimo.2019070
##### References:
 [1] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance: Mathematics and Economics, 61 (2011), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [2] T. Bielecki and M. Rutkowski, Credit Risk: Modelling, Valuation and Hedging, 2nd edition, Springer-Verlag, New York, 2002. [3] T. Bielecki and I. Jang, Portfolio optimization with a defaultable security, Asia-Pacific Financial Markets, 13 (2006), 113-127. [4] C. Blanchet-Scalliet and M. Jeanblanc, Hazard rate for credit risk and hedging defaultable contingent claims, Finance and Stochastics, 8 (2004), 145-159.  doi: 10.1007/s00780-003-0108-1. [5] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [6] L. Bo, Y. Wang and X. Yang, Stochastic portfolio optimization with default risk, Journal of Mathematical Analysis and Applications, 397 (2013), 467-480.  doi: 10.1016/j.jmaa.2012.07.058. [7] L. Chen and H. Yang, Optimal reinsurance and investment strategy with two piece utility function, Journal of Industrial and Management Optimization, 13 (2017), 737-755.  doi: 10.3934/jimo.2016044. [8] M. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Mathematics of Operations Research, 36 (2011), 604-619.  doi: 10.1287/moor.1110.0508. [9] C. Deng, X. Zeng and H. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, European Journal of Operational Research, 264 (2018), 1144-1158.  doi: 10.1016/j.ejor.2017.06.065. [10] D. Duffie and K. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies, 12 (1999), 687-720. [11] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, Journal of Computational and Applied Mathematics, 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032. [12] M. Jeanblanc and M. Rutkowski, Default risk and hazard process, in Mathematical Finance–Bachelier Congress 2000, Springer Finance, Springer, Berlin, 2002, 281–312. [13] X. Liang and L. Bai, Minimizing expected time to reach a given capital level before ruin, Journal of Industrial and Management Optimization, 13 (2017), 1771-1791.  doi: 10.3934/jimo.2017018. [14] Z. Liang, L. Bai and J. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Applications and Methods, 32 (2011), 587-608.  doi: 10.1002/oca.965. [15] Z. Liang, K. Yuen and K. Cheung, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Applied Stochastic Models in Business and Industry, 28 (2012), 585-597.  doi: 10.1002/asmb.934. [16] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002. [17] D. Landriault, B. Li, D. Li and D. Li, A pair of optimal reinsurance-investment strategies in the two-sided exit framework, Insurance: Mathematics and Economics, 71 (2016), 284-294.  doi: 10.1016/j.insmatheco.2016.09.002. [18] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for a mean-variance insurer under stochastic interest rate model and inflation risk, Insurance Mathematics and Economics, 64 (2015), 28-44.  doi: 10.1016/j.insmatheco.2015.05.003. [19] S. Li, Z. Jin, P. Chen and N. Zhang, Markowitz's mean-variance optimization with investment and constrained reinsurance, Journal of Industrial Management Optimization, 13 (2017), 375-397.  doi: 10.3934/jimo.2016022. [20] T. Pang and A. Hussain, A stochastic portfolio optimization model with complete memory, Stochastic Analysis and Applications, 35 (2017), 742-766.  doi: 10.1080/07362994.2017.1299629. [21] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance: Mathematics and Economics, 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004. [22] Y. Shen, Q. Meng and P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica, 50 (2014), 1565-1579.  doi: 10.1016/j.automatica.2014.03.021. [23] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173. [24] L. Xu, R. Wang and D. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815.  doi: 10.3934/jimo.2008.4.801. [25] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [26] Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance: Mathematics and Economics, 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001. [27] H. Zhao, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, Journal of Mathematical Analysis and Applications, 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035. [28] H. Zhu, C. Deng, S. Yue and Y. Deng, Optimal reinsurance and investment problem for an insurer with counterparty risk, Insurance: Mathematics and Economics, 61 (2015), 242-254.  doi: 10.1016/j.insmatheco.2015.01.013. [29] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance: Mathematics and Economics, 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.

show all references

##### References:
 [1] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance: Mathematics and Economics, 61 (2011), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [2] T. Bielecki and M. Rutkowski, Credit Risk: Modelling, Valuation and Hedging, 2nd edition, Springer-Verlag, New York, 2002. [3] T. Bielecki and I. Jang, Portfolio optimization with a defaultable security, Asia-Pacific Financial Markets, 13 (2006), 113-127. [4] C. Blanchet-Scalliet and M. Jeanblanc, Hazard rate for credit risk and hedging defaultable contingent claims, Finance and Stochastics, 8 (2004), 145-159.  doi: 10.1007/s00780-003-0108-1. [5] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [6] L. Bo, Y. Wang and X. Yang, Stochastic portfolio optimization with default risk, Journal of Mathematical Analysis and Applications, 397 (2013), 467-480.  doi: 10.1016/j.jmaa.2012.07.058. [7] L. Chen and H. Yang, Optimal reinsurance and investment strategy with two piece utility function, Journal of Industrial and Management Optimization, 13 (2017), 737-755.  doi: 10.3934/jimo.2016044. [8] M. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Mathematics of Operations Research, 36 (2011), 604-619.  doi: 10.1287/moor.1110.0508. [9] C. Deng, X. Zeng and H. Zhu, Non-zero-sum stochastic differential reinsurance and investment games with default risk, European Journal of Operational Research, 264 (2018), 1144-1158.  doi: 10.1016/j.ejor.2017.06.065. [10] D. Duffie and K. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies, 12 (1999), 687-720. [11] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, Journal of Computational and Applied Mathematics, 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032. [12] M. Jeanblanc and M. Rutkowski, Default risk and hazard process, in Mathematical Finance–Bachelier Congress 2000, Springer Finance, Springer, Berlin, 2002, 281–312. [13] X. Liang and L. Bai, Minimizing expected time to reach a given capital level before ruin, Journal of Industrial and Management Optimization, 13 (2017), 1771-1791.  doi: 10.3934/jimo.2017018. [14] Z. Liang, L. Bai and J. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Applications and Methods, 32 (2011), 587-608.  doi: 10.1002/oca.965. [15] Z. Liang, K. Yuen and K. Cheung, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model, Applied Stochastic Models in Business and Industry, 28 (2012), 585-597.  doi: 10.1002/asmb.934. [16] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002. [17] D. Landriault, B. Li, D. Li and D. Li, A pair of optimal reinsurance-investment strategies in the two-sided exit framework, Insurance: Mathematics and Economics, 71 (2016), 284-294.  doi: 10.1016/j.insmatheco.2016.09.002. [18] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for a mean-variance insurer under stochastic interest rate model and inflation risk, Insurance Mathematics and Economics, 64 (2015), 28-44.  doi: 10.1016/j.insmatheco.2015.05.003. [19] S. Li, Z. Jin, P. Chen and N. Zhang, Markowitz's mean-variance optimization with investment and constrained reinsurance, Journal of Industrial Management Optimization, 13 (2017), 375-397.  doi: 10.3934/jimo.2016022. [20] T. Pang and A. Hussain, A stochastic portfolio optimization model with complete memory, Stochastic Analysis and Applications, 35 (2017), 742-766.  doi: 10.1080/07362994.2017.1299629. [21] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance: Mathematics and Economics, 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004. [22] Y. Shen, Q. Meng and P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica, 50 (2014), 1565-1579.  doi: 10.1016/j.automatica.2014.03.021. [23] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173. [24] L. Xu, R. Wang and D. Yao, On maximizing the expected terminal utility by investment and reinsurance, Journal of Industrial and Management Optimization, 4 (2008), 801-815.  doi: 10.3934/jimo.2008.4.801. [25] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [26] Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance: Mathematics and Economics, 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001. [27] H. Zhao, Y. Shen and Y. Zeng, Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security, Journal of Mathematical Analysis and Applications, 437 (2016), 1036-1057.  doi: 10.1016/j.jmaa.2016.01.035. [28] H. Zhu, C. Deng, S. Yue and Y. Deng, Optimal reinsurance and investment problem for an insurer with counterparty risk, Insurance: Mathematics and Economics, 61 (2015), 242-254.  doi: 10.1016/j.insmatheco.2015.01.013. [29] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance: Mathematics and Economics, 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.
Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal investment strategy $k^{*}(t)$
Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal investment strategy $\gamma^{*}(t)$
Effect of delay parameters $u$, $\alpha$ and $\beta$ on the optimal risk control $l^{*}(t)$
Value functions with respect to $x$
Effect of delay parameters $\beta$ on the pre-default value function
Effect of the default parameters $1/\Delta$ and $\zeta$ on the pre-default value function
Model parameter values
 Symbol Value Symbol Value $\alpha$ $0.1$ $\nu$ $1$ $u$ $5$ $\lambda$ $0.3$ $\beta$ $0.3$ $\theta$ $0.1$ $r$ $0.05$ $\eta$ $0.4$ $\zeta$ $0.5$ $p$ $1$ $\Delta$ $0.25$ $c$ $0.5$ $\mu$ $0.15$ $\sigma$ $0.2$
 Symbol Value Symbol Value $\alpha$ $0.1$ $\nu$ $1$ $u$ $5$ $\lambda$ $0.3$ $\beta$ $0.3$ $\theta$ $0.1$ $r$ $0.05$ $\eta$ $0.4$ $\zeta$ $0.5$ $p$ $1$ $\Delta$ $0.25$ $c$ $0.5$ $\mu$ $0.15$ $\sigma$ $0.2$
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