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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

* Corresponding author

Received  July 2017 Revised  June 2018 Published  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.

Citation: Jiaqin Wei, Danping Li, Yan Zeng. Robust optimal consumption-investment strategy with non-exponential discounting. Journal of Industrial & Management Optimization, 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. [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.

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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