Figure 1.
Optimal investment policies with respect to the wealth and the model ambiguity
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A non-zero-sum reinsurance-investment game with delay and asymmetric information
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 |
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 [
Keywords:
A standardized penalty,
ambiguity derived ratio,
HJB equation,
model ambiguity,
optimal investment,
probability of lifetime ruin.
Mathematics Subject Classification:
Primary: 91B30; Secondary: 91B42.
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. |
show all references
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. |
Figure 5.
Ambiguity Derived Ratio with respect to model ambiguity
Figure 2.
Optimal investment policies with respect to the lifetime
Figure 3.
The value function with respect to model ambiguity
Figure 4.
Return rate of the risky asset under robust risk measure ($ \mu = 0.1 $ )
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