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doi: 10.3934/jimo.2021121
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## Robust optimal asset-liability management with penalization on ambiguity

 School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, China

* Corresponding author: Hui Mi

Received  May 2020 Revised  February 2021 Early access July 2021

Fund Project: The research was supported by National Natural Science Foundation of China (Grant No.61304065) and the Qing Lan Project of Jiangsu Province

In this paper, we study the robust optimal asset- problems for an ambiguity-averse investor, who does not have perfect information in the drift terms of the risky asset and liability processes. Two different kinds of objectives are considered: $(i)$ Maximizing the minimal expected utility of the terminal wealth; $(ii)$ Minimizing the maximal cumulative deviation. The ambiguity in both problems is described by a set of equivalent measures to the reference model. By the stochastic dynamic programming approach and Hamilton-Jacobi-Bellman (HJB) equation, we derive closed-form expressions for the value function and corresponding robust optimal investment strategy in each problem. Furthermore, some special cases are provided to investigate the effect of model uncertainty on the optimal investment strategy. Finally, the economic implication and parameter sensitivity are analyzed by some numerical examples. We also compare the robust optimal investment strategies in two different problems.

Citation: Yu Yuan, Hui Mi. Robust optimal asset-liability management with penalization on ambiguity. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021121
##### References:
 [1] E. L. Anderson, L. Hansen and T. J. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123. [2] N. Branger and L. S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, Journal of Banking and Finance, 37 (2013), 5036-5047. [3] H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182. [4] M. C. Chiu and D. Li, Asset and liability management under a continuous-time mean-variance optimization framework, Insurance Math. Econom., 39 (2006), 330-355.  doi: 10.1016/j.insmatheco.2006.03.006. [5] M. C. Chiu and H. Y. Wong, Mean-variance asset-liability management: Cointegrated assets and insurance liability, European J. Oper. Res., 223 (2012), 785-793.  doi: 10.1016/j.ejor.2012.07.009. [6] N. Gülpinar and D. Pachamanova, A robust optimization approach to asset-liability management under time-varying investment opportunities, Journal of Banking & Finance, 37 (2013), 2031-2041. [7] L. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66. [8] Y. Huang, X. Yang and J. Zhou, Robust optimal investment and reinsurance problem for a general insurance company under Heston model, Math. Methods Oper. Res., 85 (2017), 305-326.  doi: 10.1007/s00186-017-0570-8. [9] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal investment decisions with a liability: The case of defined benefit pension plans, Insurance Math. Econom., 39 (2006), 81-98.  doi: 10.1016/j.insmatheco.2006.01.005. [10] D. Li, Y. Shen and Y. Zeng, Dynamic derivative-based investment strategy for mean-variance asset-liability management with stochastic volatility, Insurance Math. Econom., 78 (2018), 72-86.  doi: 10.1016/j.insmatheco.2017.11.006. [11] D. Li, Y. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679. [12] K. Luo, G. Wang and Y. Hu, Optimal portfolio on tracking the expected wealth process with liquidity constraints, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 483-490.  doi: 10.1016/S0252-9602(11)60249-X. [13] S. Luo, M. Wang and W. Zhu, Maximizing a robust goal-reaching probability with penalization on ambiguity, J. Comput. Appl. Math., 348 (2019), 261-281.  doi: 10.1016/j.cam.2018.08.049. [14] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983. [15] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [16] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics, 80 (2008), 317-337.  doi: 10.1080/17442500701655408. [17] J. Pan, S. Hu and X. Zhou, Optimal investment strategy for asset-liability management under the Heston model, Optimization, 68 (2019), 895-920.  doi: 10.1080/02331934.2018.1561691. [18] J. Pan and Q. Xiao, Optimal mean-variance asset-liability management with stochastic interest rates and inflation risks, Math. Methods Oper. Res., 85 (2017), 491-519.  doi: 10.1007/s00186-017-0580-6. [19] J. Pan and Q. Xiao, Optimal asset-liability management with liquidity constraints and stochastic interest rates in the expected utility framework, J. Comput. Appl. Math., 317 (2017), 371-387.  doi: 10.1016/j.cam.2016.11.037. [20] C. S. Pun and H. Y. Wong, Robust investment-reinsurance optimization with multiscale stochastic volatility, Insurance Math. Econom., 62 (2015), 245-256.  doi: 10.1016/j.insmatheco.2015.03.030. [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. [22] 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. [23] B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scand. Actuar. J., 2015 (2015), 725-751.  doi: 10.1080/03461238.2014.883085. [24] 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. [25] Y. Zeng, Z. Li and H. Wu, Optimal portfolio selection in a L$\acute{e}$vy market with uncontrolled cash flow and only risky assets, Internat. J. Control, 86 (2013), 426-437.  doi: 10.1080/00207179.2012.735373.

show all references

##### References:
 [1] E. L. Anderson, L. Hansen and T. J. Sargent, A quartet of semigroups for model specification, robustness, prices of risk, and model detection, Journal of the European Economic Association, 1 (2003), 68-123. [2] N. Branger and L. S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, Journal of Banking and Finance, 37 (2013), 5036-5047. [3] H. Chang, Dynamic mean-variance portfolio selection with liability and stochastic interest rate, Economic Modelling, 51 (2015), 172-182. [4] M. C. Chiu and D. Li, Asset and liability management under a continuous-time mean-variance optimization framework, Insurance Math. Econom., 39 (2006), 330-355.  doi: 10.1016/j.insmatheco.2006.03.006. [5] M. C. Chiu and H. Y. Wong, Mean-variance asset-liability management: Cointegrated assets and insurance liability, European J. Oper. Res., 223 (2012), 785-793.  doi: 10.1016/j.ejor.2012.07.009. [6] N. Gülpinar and D. Pachamanova, A robust optimization approach to asset-liability management under time-varying investment opportunities, Journal of Banking & Finance, 37 (2013), 2031-2041. [7] L. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66. [8] Y. Huang, X. Yang and J. Zhou, Robust optimal investment and reinsurance problem for a general insurance company under Heston model, Math. Methods Oper. Res., 85 (2017), 305-326.  doi: 10.1007/s00186-017-0570-8. [9] R. Josa-Fombellida and J. P. Rincón-Zapatero, Optimal investment decisions with a liability: The case of defined benefit pension plans, Insurance Math. Econom., 39 (2006), 81-98.  doi: 10.1016/j.insmatheco.2006.01.005. [10] D. Li, Y. Shen and Y. Zeng, Dynamic derivative-based investment strategy for mean-variance asset-liability management with stochastic volatility, Insurance Math. Econom., 78 (2018), 72-86.  doi: 10.1016/j.insmatheco.2017.11.006. [11] D. Li, Y. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679. [12] K. Luo, G. Wang and Y. Hu, Optimal portfolio on tracking the expected wealth process with liquidity constraints, Acta Math. Sci. Ser. B (Engl. Ed.), 31 (2011), 483-490.  doi: 10.1016/S0252-9602(11)60249-X. [13] S. Luo, M. Wang and W. Zhu, Maximizing a robust goal-reaching probability with penalization on ambiguity, J. Comput. Appl. Math., 348 (2019), 261-281.  doi: 10.1016/j.cam.2018.08.049. [14] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983. [15] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [16] S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics, 80 (2008), 317-337.  doi: 10.1080/17442500701655408. [17] J. Pan, S. Hu and X. Zhou, Optimal investment strategy for asset-liability management under the Heston model, Optimization, 68 (2019), 895-920.  doi: 10.1080/02331934.2018.1561691. [18] J. Pan and Q. Xiao, Optimal mean-variance asset-liability management with stochastic interest rates and inflation risks, Math. Methods Oper. Res., 85 (2017), 491-519.  doi: 10.1007/s00186-017-0580-6. [19] J. Pan and Q. Xiao, Optimal asset-liability management with liquidity constraints and stochastic interest rates in the expected utility framework, J. Comput. Appl. Math., 317 (2017), 371-387.  doi: 10.1016/j.cam.2016.11.037. [20] C. S. Pun and H. Y. Wong, Robust investment-reinsurance optimization with multiscale stochastic volatility, Insurance Math. Econom., 62 (2015), 245-256.  doi: 10.1016/j.insmatheco.2015.03.030. [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. [22] 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. [23] B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scand. Actuar. J., 2015 (2015), 725-751.  doi: 10.1080/03461238.2014.883085. [24] 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. [25] Y. Zeng, Z. Li and H. Wu, Optimal portfolio selection in a L$\acute{e}$vy market with uncontrolled cash flow and only risky assets, Internat. J. Control, 86 (2013), 426-437.  doi: 10.1080/00207179.2012.735373.
The influence of $\mu$ and $\rho$ on the robust optimal investment strategy
The influence of $b$ and $\beta_1$ on the robust optimal investment strategy
The influence of $R$ and $y_0$ on the robust optimal investment strategy
The basic parameters
 r μ σ a b ρ t T m β1 β2 x R y0 0.03 0.08 0.25 0.1 0.2 0.3 0 10 0.5 0.5 0.5 1 0.1 1
 r μ σ a b ρ t T m β1 β2 x R y0 0.03 0.08 0.25 0.1 0.2 0.3 0 10 0.5 0.5 0.5 1 0.1 1
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