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2020, 16(1): 207-230.
doi: 10.3934/jimo.2018147
Robust optimal consumption-investment strategy with non-exponential discounting
| 1. | School of Statistics, East China Normal University, Shanghai, 200241, China |
| 2. | Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, China |
This paper extends the existing dynamic consumption-investment problem to the case with more general discount functions under the robust framework. The decision-maker is ambiguity-averse and invests her wealth in a risk-free asset and a risky asset. Since non-exponential discounting is considered in our model, our optimization problem is time inconsistent. By solving the extended Hamilton-Jacobi-Bellman equations, the corresponding optimal consumption-investment strategies for sophisticated and naive investors under power and logarithmic utility functions are derived explicitly. Our model and results extend some existing ones and derive some interesting phenomena.
Keywords:
Non-exponential discounting,
equilibrium strategy,
optimal consumption-investment problem,
Hamilton-Jacobi-Bellman equation.
Mathematics Subject Classification:
Primary: 90B50, 93E20; Secondary: 91G80.
Citation:
Jiaqin Wei, Danping Li, Yan Zeng. Robust optimal consumption-investment strategy with non-exponential discounting. Journal of Industrial and Management Optimization,
2020, 16
(1)
: 207-230.
doi: 10.3934/jimo.2018147
![]()
References:
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A. Altarovici, J. Muhle-Karbe and H. M. Soner,
Asymptotics for fixed transaction costs, Finance and Stochastics, 19 (2015), 363-414.
doi: 10.1007/s00780-015-0261-3. |
| [2] |
E. W. 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.
doi: 10.4324/9780203358061_chapter_16. |
| [3] |
D. Bertsimas and V. Goyal,
On the approximability of adjustable robust convex optimization under uncertainty, Mathematical Methods of Operations Research, 77 (2013), 323-343.
doi: 10.1007/s00186-012-0405-6. |
| [4] |
B. Berdjane and S. Pergamenshchikov,
Optimal consumption and investment for markets with random coefficients, Finance and Stochastics, 17 (2013), 419-446.
doi: 10.1007/s00780-012-0193-0. |
| [5] |
T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, 2010, Working Paper, Stockholm School of Economics. |
| [6] |
T. Bjork, A. Murgoci and X.Y. Zhou,
Mean-variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.
doi: 10.1111/j.1467-9965.2011.00515.x. |
| [7] |
N. Castanedaleyva and D. Hernandezhernandez,
Optimal consumption-investment problems in incomplete markets with stochastic coefficients, SIAM Journal on Control and Optimization, 44 (2005), 1322-1344.
doi: 10.1137/S0363012904440885. |
| [8] |
S. Chen, Z. Li and Y. Zeng,
Optimal dividend strategies with time-inconsistent preferences, Journal of Economic Dynamics and Control, 46 (2014), 150-172.
doi: 10.1016/j.jedc.2014.06.018. |
| [9] |
J. F. Cocco, F. Gomes and P. J. Maenhout,
Consumption and portfolio choice over the life cycle, Review of Financial Studies, 18 (2005), 491-533.
doi: 10.1093/rfs/hhi017. |
| [10] |
X. Cui, D. Li and X. Li,
Mean-variance policy for discrete-time cone-constrained markets: Time consistency in efficiency and the minimum-variance signed supermartingale meansure, Mathematical Finance, 27 (2017), 471-504.
doi: 10.1111/mafi.12093. |
| [11] |
X. Cui, D. Li and Y. Shi,
Self-coordination in time inconsistent stochastic decision problems: A planner-doer game framework, Journal of Economic Dynamics and Control, 75 (2017), 91-113.
doi: 10.1016/j.jedc.2016.12.001. |
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C. Czichowsky,
Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance and Stochastics, 17 (2013), 227-271.
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L. Delong and C. Klüppelberg,
Optimal investment and consumption in a Black-Scholes market with Lévy-driven stochastic coefficients, Annals of Applied Probability, 18 (2008), 879-908.
doi: 10.1214/07-AAP475. |
| [14] |
I. Ekeland and A. Lazrak,
The golden rule when preferences are time inconsistent, Mathematics and Financial Economics, 4 (2010), 29-55.
doi: 10.1007/s11579-010-0034-x. |
| [15] |
I. Ekeland and T. A. Pirvu,
Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.
doi: 10.1007/s11579-008-0014-6. |
| [16] |
I. Ekeland, O. Mbodji and T. A. Pirvu,
Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32.
doi: 10.1137/100810034. |
| [17] |
W. H. Fleming and D. Hernandezhernandez,
An optimal consumption model with stochastic volatility, Finance and Stochastics, 7 (2003), 245-262.
doi: 10.1007/s007800200083. |
| [18] |
C. R. Flor and L. S. Larsen,
Robust portfolio choice with stochastic interest rates, Annals of
Finance, 10 (2014), 243-265.
doi: 10.1007/s10436-013-0234-5. |
| [19] |
S. Frederick, G. Loewenstein and T. Odonoghue,
Time discounting and time preference: A critical review, Journal of Economic Literature, 40 (2002), 351-401.
doi: 10.1257/jel.40.2.351. |
| [20] |
N. Gülpinar and E. Çanakoḡlu,
Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2016), 500-523.
|
| [21] |
Y. Hu, H. Jin and X. Zhou,
Time-inconsistent stochastic linear-quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.
doi: 10.1137/110853960. |
| [22] |
J. Kallsen and J. Muhlekarbe,
The general structure of optimal investment and consumption with small transaction costs, Mathematical Finance, 27 (2017), 659-703.
doi: 10.1111/mafi.12106. |
| [23] |
L. Karp,
Non-constant discounting in continuous time, Journal of Economic Theory, 132 (2007), 557-568.
doi: 10.1016/j.jet.2005.07.006. |
| [24] |
H. Liu,
Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454.
|
| [25] |
H. Liu,
Dynamic portfolio choice under ambiguity and regime switching mean returns, Journal of Economic Dynamics and Control, 35 (2011), 623-640.
doi: 10.1016/j.jedc.2010.12.012. |
| [26] |
G. Loewenstein and D. Prelec,
Anomalies in intertemporal choice: Evidence and an interpretation, Quarterly Journal of Economics, 107 (1992), 573-597.
doi: 10.2307/2118482. |
| [27] |
P. J. Maenhout,
Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.
doi: 10.1093/rfs/hhh003. |
| [28] |
P. J. Maenhout,
Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium, Journal of Economic Theory, 128 (2006), 136-163.
doi: 10.1016/j.jet.2005.12.012. |
| [29] |
A. Matoussi, D. Possamaï and C. Zhou,
Robust utility maximization in nondominated models with 2BSDE: The uncerain volatility model, Mathematical Finance, 25 (2015), 258-287.
doi: 10.1111/mafi.12031. |
| [30] |
J. Marín-Solano and J. Navas,
Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872.
doi: 10.1016/j.ejor.2009.04.005. |
| [31] |
R. C. Merton,
Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.
doi: 10.2307/1926560. |
| [32] |
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. |
| [33] |
C. Munk and A. N. Rubtsov,
Portfolio management with stochastic interest rates and inflation ambiguity, Annals of Finance, 10 (2014), 419-455.
doi: 10.1007/s10436-013-0238-1. |
| [34] |
B. Øksendal,
Stochastic Differential Equations: An Introduction with Applications Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-13050-6. |
| [35] |
R. A. Pollak,
Consistent planning, Review of Financial Studies, 35 (1968), 201-208.
doi: 10.2307/2296548. |
| [36] |
A. Schied,
Robust optimal control for a consumption-investment problem, Mathematical Methods of Operations Research, 67 (2008), 1-20.
doi: 10.1007/s00186-007-0172-y. |
| [37] |
Y. Shen and J. Wei,
Optimal investment-consumption-insurance with random parameters, Scandinavian Actuarial Journal, 2016 (2016), 37-62.
doi: 10.1080/03461238.2014.900518. |
| [38] |
R. Strotz,
Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128-143.
doi: 10.1007/978-1-349-15492-0_10. |
| [39] |
Z. Sun, X. Zheng and X. Zhang,
Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.
doi: 10.1016/j.jmaa.2016.09.053. |
| [40] |
J. Yong,
Time-inconsistent optimal control problems and the equilibrium {HJB} equation, Mathematical Control and Related Fields, 2 (2012), 271-329.
doi: 10.3934/mcrf.2012.2.271. |
| [41] |
X. Zeng and M. Taksar,
A stochastic volatility model and optimal portfolio selection, Quantitative Finance, 13 (2013), 1547-1558.
doi: 10.1080/14697688.2012.740568. |
| [42] |
Y. Zeng, D. Li and A. Gu,
Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance: Mathematics and Economics, 66 (2016), 138-152.
doi: 10.1016/j.insmatheco.2015.10.012. |
| [43] |
Y. Zeng, D. Li, Z. Chen and Z. Yang,
Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility, Journal of Economic Dynamic and Control, 88 (2018), 70-103.
doi: 10.1016/j.jedc.2018.01.023. |
| [44] |
Q. Zhao, Y. Shen and J. Wei,
Exponential utility maximization for an insurer with time-inconsistent preferences, Insurance: Mathematics and Economics, 70 (2016), 89-104.
doi: 10.1016/j.insmatheco.2016.06.003. |
| [45] |
Q. Zhao, R. Wang and J. Wei,
Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835.
doi: 10.1016/j.ejor.2014.04.034. |
show all references
References:
| [1] |
A. Altarovici, J. Muhle-Karbe and H. M. Soner,
Asymptotics for fixed transaction costs, Finance and Stochastics, 19 (2015), 363-414.
doi: 10.1007/s00780-015-0261-3. |
| [2] |
E. W. 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.
doi: 10.4324/9780203358061_chapter_16. |
| [3] |
D. Bertsimas and V. Goyal,
On the approximability of adjustable robust convex optimization under uncertainty, Mathematical Methods of Operations Research, 77 (2013), 323-343.
doi: 10.1007/s00186-012-0405-6. |
| [4] |
B. Berdjane and S. Pergamenshchikov,
Optimal consumption and investment for markets with random coefficients, Finance and Stochastics, 17 (2013), 419-446.
doi: 10.1007/s00780-012-0193-0. |
| [5] |
T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, 2010, Working Paper, Stockholm School of Economics. |
| [6] |
T. Bjork, A. Murgoci and X.Y. Zhou,
Mean-variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.
doi: 10.1111/j.1467-9965.2011.00515.x. |
| [7] |
N. Castanedaleyva and D. Hernandezhernandez,
Optimal consumption-investment problems in incomplete markets with stochastic coefficients, SIAM Journal on Control and Optimization, 44 (2005), 1322-1344.
doi: 10.1137/S0363012904440885. |
| [8] |
S. Chen, Z. Li and Y. Zeng,
Optimal dividend strategies with time-inconsistent preferences, Journal of Economic Dynamics and Control, 46 (2014), 150-172.
doi: 10.1016/j.jedc.2014.06.018. |
| [9] |
J. F. Cocco, F. Gomes and P. J. Maenhout,
Consumption and portfolio choice over the life cycle, Review of Financial Studies, 18 (2005), 491-533.
doi: 10.1093/rfs/hhi017. |
| [10] |
X. Cui, D. Li and X. Li,
Mean-variance policy for discrete-time cone-constrained markets: Time consistency in efficiency and the minimum-variance signed supermartingale meansure, Mathematical Finance, 27 (2017), 471-504.
doi: 10.1111/mafi.12093. |
| [11] |
X. Cui, D. Li and Y. Shi,
Self-coordination in time inconsistent stochastic decision problems: A planner-doer game framework, Journal of Economic Dynamics and Control, 75 (2017), 91-113.
doi: 10.1016/j.jedc.2016.12.001. |
| [12] |
C. Czichowsky,
Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance and Stochastics, 17 (2013), 227-271.
doi: 10.1007/s00780-012-0189-9. |
| [13] |
L. Delong and C. Klüppelberg,
Optimal investment and consumption in a Black-Scholes market with Lévy-driven stochastic coefficients, Annals of Applied Probability, 18 (2008), 879-908.
doi: 10.1214/07-AAP475. |
| [14] |
I. Ekeland and A. Lazrak,
The golden rule when preferences are time inconsistent, Mathematics and Financial Economics, 4 (2010), 29-55.
doi: 10.1007/s11579-010-0034-x. |
| [15] |
I. Ekeland and T. A. Pirvu,
Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.
doi: 10.1007/s11579-008-0014-6. |
| [16] |
I. Ekeland, O. Mbodji and T. A. Pirvu,
Time-consistent portfolio management, SIAM Journal on Financial Mathematics, 3 (2012), 1-32.
doi: 10.1137/100810034. |
| [17] |
W. H. Fleming and D. Hernandezhernandez,
An optimal consumption model with stochastic volatility, Finance and Stochastics, 7 (2003), 245-262.
doi: 10.1007/s007800200083. |
| [18] |
C. R. Flor and L. S. Larsen,
Robust portfolio choice with stochastic interest rates, Annals of
Finance, 10 (2014), 243-265.
doi: 10.1007/s10436-013-0234-5. |
| [19] |
S. Frederick, G. Loewenstein and T. Odonoghue,
Time discounting and time preference: A critical review, Journal of Economic Literature, 40 (2002), 351-401.
doi: 10.1257/jel.40.2.351. |
| [20] |
N. Gülpinar and E. Çanakoḡlu,
Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2016), 500-523.
|
| [21] |
Y. Hu, H. Jin and X. Zhou,
Time-inconsistent stochastic linear-quadratic control, SIAM Journal on Control and Optimization, 50 (2012), 1548-1572.
doi: 10.1137/110853960. |
| [22] |
J. Kallsen and J. Muhlekarbe,
The general structure of optimal investment and consumption with small transaction costs, Mathematical Finance, 27 (2017), 659-703.
doi: 10.1111/mafi.12106. |
| [23] |
L. Karp,
Non-constant discounting in continuous time, Journal of Economic Theory, 132 (2007), 557-568.
doi: 10.1016/j.jet.2005.07.006. |
| [24] |
H. Liu,
Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454.
|
| [25] |
H. Liu,
Dynamic portfolio choice under ambiguity and regime switching mean returns, Journal of Economic Dynamics and Control, 35 (2011), 623-640.
doi: 10.1016/j.jedc.2010.12.012. |
| [26] |
G. Loewenstein and D. Prelec,
Anomalies in intertemporal choice: Evidence and an interpretation, Quarterly Journal of Economics, 107 (1992), 573-597.
doi: 10.2307/2118482. |
| [27] |
P. J. Maenhout,
Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.
doi: 10.1093/rfs/hhh003. |
| [28] |
P. J. Maenhout,
Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium, Journal of Economic Theory, 128 (2006), 136-163.
doi: 10.1016/j.jet.2005.12.012. |
| [29] |
A. Matoussi, D. Possamaï and C. Zhou,
Robust utility maximization in nondominated models with 2BSDE: The uncerain volatility model, Mathematical Finance, 25 (2015), 258-287.
doi: 10.1111/mafi.12031. |
| [30] |
J. Marín-Solano and J. Navas,
Consumption and portfolio rules for time-inconsistent investors, European Journal of Operational Research, 201 (2010), 860-872.
doi: 10.1016/j.ejor.2009.04.005. |
| [31] |
R. C. Merton,
Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.
doi: 10.2307/1926560. |
| [32] |
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. |
| [33] |
C. Munk and A. N. Rubtsov,
Portfolio management with stochastic interest rates and inflation ambiguity, Annals of Finance, 10 (2014), 419-455.
doi: 10.1007/s10436-013-0238-1. |
| [34] |
B. Øksendal,
Stochastic Differential Equations: An Introduction with Applications Springer-Verlag, Berlin, 1985.
doi: 10.1007/978-3-662-13050-6. |
| [35] |
R. A. Pollak,
Consistent planning, Review of Financial Studies, 35 (1968), 201-208.
doi: 10.2307/2296548. |
| [36] |
A. Schied,
Robust optimal control for a consumption-investment problem, Mathematical Methods of Operations Research, 67 (2008), 1-20.
doi: 10.1007/s00186-007-0172-y. |
| [37] |
Y. Shen and J. Wei,
Optimal investment-consumption-insurance with random parameters, Scandinavian Actuarial Journal, 2016 (2016), 37-62.
doi: 10.1080/03461238.2014.900518. |
| [38] |
R. Strotz,
Myopia and inconsistency in dynamic utility maximization, Readings in Welfare Economics, (1973), 128-143.
doi: 10.1007/978-1-349-15492-0_10. |
| [39] |
Z. Sun, X. Zheng and X. Zhang,
Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.
doi: 10.1016/j.jmaa.2016.09.053. |
| [40] |
J. Yong,
Time-inconsistent optimal control problems and the equilibrium {HJB} equation, Mathematical Control and Related Fields, 2 (2012), 271-329.
doi: 10.3934/mcrf.2012.2.271. |
| [41] |
X. Zeng and M. Taksar,
A stochastic volatility model and optimal portfolio selection, Quantitative Finance, 13 (2013), 1547-1558.
doi: 10.1080/14697688.2012.740568. |
| [42] |
Y. Zeng, D. Li and A. Gu,
Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance: Mathematics and Economics, 66 (2016), 138-152.
doi: 10.1016/j.insmatheco.2015.10.012. |
| [43] |
Y. Zeng, D. Li, Z. Chen and Z. Yang,
Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility, Journal of Economic Dynamic and Control, 88 (2018), 70-103.
doi: 10.1016/j.jedc.2018.01.023. |
| [44] |
Q. Zhao, Y. Shen and J. Wei,
Exponential utility maximization for an insurer with time-inconsistent preferences, Insurance: Mathematics and Economics, 70 (2016), 89-104.
doi: 10.1016/j.insmatheco.2016.06.003. |
| [45] |
Q. Zhao, R. Wang and J. Wei,
Consumption-investment strategies with non-exponential discounting and logarithmic utility, European Journal of Operational Research, 238 (2014), 824-835.
doi: 10.1016/j.ejor.2014.04.034. |
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