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January  2020, 16(1): 207-230. doi: 10.3934/jimo.2018147

Robust optimal consumption-investment strategy with non-exponential discounting

Jiaqin Wei 1, , Danping Li 1, and  Yan Zeng 2,,

1. 

School of Statistics, East China Normal University, Shanghai, 200241, China
2. 

Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, China

* Corresponding author

Received  July 2017 Revised  June 2018 Published  January 2020 Early access  September 2018

Fund Project: This work was supported by Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019), National Natural Science Foundation of China (11771466, 11601157, 11231005, 71571195, 71771220, 71721001, 11801179), Major Program of the National Social Science Foundation of China (No. 17ZDA073), Fok Ying Tung Education Foundation for Young Teachers in the Higher Education Institutions of China (151081), Guangdong Natural Science Foundation for Research Team (2014A030312003), Guangdong Natural Science Foundation for Distinguished Young Scholar (2015A030306040), and Fundamental Research Funds for the Central Universities (No. 17wkzd08).

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 & Management Optimization, 2020, 16 (1) : 207-230. doi: 10.3934/jimo.2018147
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, 2010, Working Paper, Stockholm School of Economics. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

N. Gülpinar and E. Çanakoḡlu, Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2016), 500-523.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-13050-6.  Google Scholar

[35]

R. A. Pollak, Consistent planning, Review of Financial Studies, 35 (1968), 201-208.  doi: 10.2307/2296548.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, 2010, Working Paper, Stockholm School of Economics. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

N. Gülpinar and E. Çanakoḡlu, Robust portfolio selection problem under temperature uncertainty, European Journal of Operational Research, 256 (2016), 500-523.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

H. Liu, Robust consumption and portfolio choice for time varying investment opportunities, Annals of Finance, 6 (2010), 435-454.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-13050-6.  Google Scholar

[35]

R. A. Pollak, Consistent planning, Review of Financial Studies, 35 (1968), 201-208.  doi: 10.2307/2296548.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Effects of $\eta$ on optimal proportion of wealth to consume and optimal value function
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