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March  2021, 17(2): 937-952. doi: 10.3934/jimo.2020005

## Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin

 1 School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China 2 China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

* Corresponding author: Ming Zhou

Received  August 2018 Revised  June 2019 Published  March 2021 Early access  January 2020

Fund Project: This research is supported by by the National Natural Science Foundation of China (11971506, 11571388), Beijing Social Science Foundation (15JGB046), the MOE Project of Key Research Institute of Humanities and Social Science at Universities (15JJD790036), and the 111 Project (B17050)

Robust portfolio selection has become a popular problem in recent years. In this paper, we study the optimal investment problem for an individual who carries a constant consumption rate but worries about the model ambiguity of the financial market. Instead of using a conventional value function such as the utility of terminal wealth maximization, here, we focus on the purpose of risk control and seek to minimize the probability of lifetime ruin. This study is motivated by the work of [3], except that we use a standardized penalty for ambiguity aversion. The reason for taking a standardized penalty is to convert the penalty to units of the value function, which makes the difference meaningful in the definition of the value function. The advantage of taking a standardized penalty is that the closed-form solutions to both the robust investment policy and the value function can be obtained. More interestingly, we use the "Ambiguity Derived Ratio" to characterize the existence of model ambiguity which significantly affects the optimal investment policy. Finally, several numerical examples are given to illustrate our results.

Citation: Bing Liu, Ming Zhou. Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin. Journal of Industrial and Management Optimization, 2021, 17 (2) : 937-952. doi: 10.3934/jimo.2020005
##### References:
 [1] E. Anderson, L. P. 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] E. Bayraktar and V. R. Young, Correspondence between lifetime minimum wealth and utility of consumption, Finance Stochastics, 11 (2007), 213-236.  doi: 10.1007/s00780-007-0035-7. [3] E. Bayraktar and Y. Zhang, Minimizing the probability of lifetime ruin under ambiguity aversion, SIAM Journal on Control and Optimization, 53 (2015), 58-90.  doi: 10.1137/140955999. [4] S. Browne, Risk-constrained dynamic active portfolio management, World Scientific Handbook in Financial Economics Series, 3 (2011), 373-354.  doi: 10.1142/9789814293501_0026. [5] W. H. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer, New York, 2006. [6] L. P. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66. [7] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400829385. [8] L. P. Hansen, T. J. Sargent, G. Turmuhambetova and N. Williams, Robust control and model misspecification, Journal of Economic Theory, 128 (2006), 45-90.  doi: 10.1016/j.jet.2004.12.006. [9] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 2 edition, Imperial College Press, 2005.  doi: 10.1142/p386. [10] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [11] H. Meng, F. L. Yuen, K. T. Siu and H. L. Yang, Optimal portfolio in a continuous-time self-exciting threshold model, Journal of Industrial and Management Optimization, 9 (2013), 487-504.  doi: 10.3934/jimo.2013.9.487. [12] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [13] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [14] S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609-692.  doi: 10.1214/aoap/1177004966. [15] L. Sun and L. H. Zhang, Optimal consumption and investment under irrational beliefs, Journal of Industrial and Management Optimization, 7 (2011), 139-156.  doi: 10.3934/jimo.2011.7.139. [16] R. Uppal and T. Wang, Model misspecification and underdiversification, The Journal of Finance, 58 (2003), 2465-2486.  doi: 10.1046/j.1540-6261.2003.00612.x. [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: Mathematics and Economics, 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011. [18] C. C. Yin and Y. Z. Wen, An extension of Paulsen-Gjessing's risk model with stochastic return on investments, Insurance: Mathematics and Economics, 52 (2013), 469-476.  doi: 10.1016/j.insmatheco.2013.02.014. [19] V. R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126.  doi: 10.1080/10920277.2004.10596174. [20] T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85.  doi: 10.1137/S0363012991218827. [21] X. Zhang, T. K. Siu and Q. B. Meng, Portfolio selection in the enlarged Markovian regime-switching market, SIAM Journal on Control and Optimization, 48 (2010), 3368-3388.  doi: 10.1137/080736351. [22] M. Zhou and K. C. Yuen, Portfolio selection by minimizing the present value of capital injection costs, Astin Bulletin, 45 (2015), 207-238.  doi: 10.1017/asb.2014.22. [23] M. Zhou, K. C. Yuen and C. C. Yin, Optimal investment and premium control for insurers with a nonlinear diffusion model, Acta Mathematicae Applicatae Sinica (English Series), 33 (2017), 945-958.  doi: 10.1007/s10255-017-0709-7.

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##### References:
 [1] E. Anderson, L. P. 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] E. Bayraktar and V. R. Young, Correspondence between lifetime minimum wealth and utility of consumption, Finance Stochastics, 11 (2007), 213-236.  doi: 10.1007/s00780-007-0035-7. [3] E. Bayraktar and Y. Zhang, Minimizing the probability of lifetime ruin under ambiguity aversion, SIAM Journal on Control and Optimization, 53 (2015), 58-90.  doi: 10.1137/140955999. [4] S. Browne, Risk-constrained dynamic active portfolio management, World Scientific Handbook in Financial Economics Series, 3 (2011), 373-354.  doi: 10.1142/9789814293501_0026. [5] W. H. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition, Springer, New York, 2006. [6] L. P. Hansen and T. J. Sargent, Robust control and model uncertainty, American Economic Review, 91 (2001), 60-66. [7] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400829385. [8] L. P. Hansen, T. J. Sargent, G. Turmuhambetova and N. Williams, Robust control and model misspecification, Journal of Economic Theory, 128 (2006), 45-90.  doi: 10.1016/j.jet.2004.12.006. [9] F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 2 edition, Imperial College Press, 2005.  doi: 10.1142/p386. [10] P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003. [11] H. Meng, F. L. Yuen, K. T. Siu and H. L. Yang, Optimal portfolio in a continuous-time self-exciting threshold model, Journal of Industrial and Management Optimization, 9 (2013), 487-504.  doi: 10.3934/jimo.2013.9.487. [12] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560. [13] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [14] S. E. Shreve and H. M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609-692.  doi: 10.1214/aoap/1177004966. [15] L. Sun and L. H. Zhang, Optimal consumption and investment under irrational beliefs, Journal of Industrial and Management Optimization, 7 (2011), 139-156.  doi: 10.3934/jimo.2011.7.139. [16] R. Uppal and T. Wang, Model misspecification and underdiversification, The Journal of Finance, 58 (2003), 2465-2486.  doi: 10.1046/j.1540-6261.2003.00612.x. [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: Mathematics and Economics, 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011. [18] C. C. Yin and Y. Z. Wen, An extension of Paulsen-Gjessing's risk model with stochastic return on investments, Insurance: Mathematics and Economics, 52 (2013), 469-476.  doi: 10.1016/j.insmatheco.2013.02.014. [19] V. R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126.  doi: 10.1080/10920277.2004.10596174. [20] T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85.  doi: 10.1137/S0363012991218827. [21] X. Zhang, T. K. Siu and Q. B. Meng, Portfolio selection in the enlarged Markovian regime-switching market, SIAM Journal on Control and Optimization, 48 (2010), 3368-3388.  doi: 10.1137/080736351. [22] M. Zhou and K. C. Yuen, Portfolio selection by minimizing the present value of capital injection costs, Astin Bulletin, 45 (2015), 207-238.  doi: 10.1017/asb.2014.22. [23] M. Zhou, K. C. Yuen and C. C. Yin, Optimal investment and premium control for insurers with a nonlinear diffusion model, Acta Mathematicae Applicatae Sinica (English Series), 33 (2017), 945-958.  doi: 10.1007/s10255-017-0709-7.
Optimal investment policies with respect to the wealth and the model ambiguity
Ambiguity Derived Ratio with respect to model ambiguity
Optimal investment policies with respect to the lifetime
The value function with respect to model ambiguity
Return rate of the risky asset under robust risk measure ($\mu = 0.1$)
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