American Institute of Mathematical Sciences

Advanced
September  2016, 6(3): 517-534. doi: 10.3934/mcrf.2016014

An optimal consumption-investment model with constraint on consumption

Zuo Quan Xu 1, and  Fahuai Yi 2,

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
2. 

School of Finance, Guangdong University of Foreign Studies, Guangzhou 510420, China

Received  June 2015 Revised  March 2016 Published  August 2016

A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investor's wealth and bankruptcy is prohibited. The investor's objective is to maximize the total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique (known) possibly exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth. According to this model, an investor should take the similar investment strategy as in Merton's model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investor's financial situation: he should use a similar consumption strategy as in Merton's model when he is in a bad situation, and consume as much as possible when he is in a good situation.
Keywords: constrained consumption., constrained viscosity solution, stochastic control in finance, free boundary problem, Optimal consumption-investment model.
Mathematics Subject Classification: Primary: 91G10; Secondary: 91G80, 49L2.
Citation: Zuo Quan Xu, Fahuai Yi. An optimal consumption-investment model with constraint on consumption. Mathematical Control & Related Fields, 2016, 6 (3) : 517-534. doi: 10.3934/mcrf.2016014
References:
[1]

M. Akian, J. L. Menaldi and A. Sulem, On an investment-consumption model with transaction costs,, SIAM Journal on Control and Optimization, 34 (1996), 329.  doi: 10.1137/S0363012993247159.  Google Scholar

[2]

I. Bardhan, Consumption and investment under constraints,, Journal of Economic Dynamics and Control, 18 (1994), 909.   Google Scholar

[3]

X. S. Chen and F. H. Yi, A problem of singular stochastic control with optimal stopping in finite horizon,, SIAM Journal on Control and Optimization, 50 (2012), 2151.  doi: 10.1137/110832264.  Google Scholar

[4]

M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. AMS, 277 (1983), 1.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[5]

J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization,, Annals of Applied Probability, 2 (1992), 767.  doi: 10.1214/aoap/1177005576.  Google Scholar

[6]

J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios,, Annals of Applied Probability, 3 (1993), 652.  doi: 10.1214/aoap/1177005357.  Google Scholar

[7]

M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity,, Mathematical Finance, 21 (2011), 775.  doi: 10.1111/j.1467-9965.2010.00449.x.  Google Scholar

[8]

M. Dai, Z. Q. Xu and X. Y. Zhou, Continuous-time mean-variance portfolio selection with proportional transaction costs,, SIAM Journal on Financial Mathematics, 1 (2010), 96.  doi: 10.1137/080742889.  Google Scholar

[9]

M. Dai and F. H. Yi, Finite horizon optimal investment with transaction costs: A parabolic double obstacle problem,, Journal of Differential Equations, 246 (2009), 1445.  doi: 10.1016/j.jde.2008.11.003.  Google Scholar

[10]

M. H. A. Davis and A. Norman, Portfolio selection with transaction costs,, Mathematics of Operations Research, 15 (1990), 676.  doi: 10.1287/moor.15.4.676.  Google Scholar

[11]

R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint,, Finance and Stochastics, 12 (2008), 299.  doi: 10.1007/s00780-008-0066-8.  Google Scholar

[12]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Second edition. Stochastic Modelling and Applied Probability, (2006).   Google Scholar

[13]

W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints,, Mathematics of Operations Research, 16 (1991), 802.  doi: 10.1287/moor.16.4.802.  Google Scholar

[14]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 2,, Communications in Partial Differential Equations, 8 (1983), 1229.  doi: 10.1080/03605308308820301.  Google Scholar

[15]

H. Markowitz, Portfolio selection,, Journal of Finance, 7 (1952), 77.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[16]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments,, John Wiley & Sons, (1959).   Google Scholar

[17]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case,, Review of Economics and Statistics, 51 (1969), 247.  doi: 10.2307/1926560.  Google Scholar

[18]

R. C. Merton, Optimum consumption and portfolio rules in a continuous time model,, Journal of Economic Theory, 3 (1971), 373.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[19]

R. C. Merton, Theory of finance from the perspective of continuous time,, Journal of Financial and Quantitative Analysis, 10 (1975), 659.  doi: 10.2307/2330617.  Google Scholar

[20]

P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming,, Review of Economics and Statistics, 51 (1969), 239.   Google Scholar

[21]

P. S. Sethi, Optimal Consumption and Investment with Bankruptcy,, Kluwer Academic Publishers, (1997).  doi: 10.1007/978-1-4615-6257-3.  Google Scholar

[22]

S. Shreve and M. Soner, Optimal investment and consumption with transaction costs,, Annals of Applied Probability, 4 (1994), 609.  doi: 10.1214/aoap/1177004966.  Google Scholar

[23]

T. Zariphopoulou, Investment-consumption models with transaction fees and Markov chain parameters,, SIAM Journal on Control and Optimization, 30 (1992), 613.  doi: 10.1137/0330035.  Google Scholar

[24]

T. Zariphopoulou, Consumption-investment models with constraints,, SIAM Journal on Control and Optimization, 32 (1994), 59.  doi: 10.1137/S0363012991218827.  Google Scholar

show all references

References:
[1]

M. Akian, J. L. Menaldi and A. Sulem, On an investment-consumption model with transaction costs,, SIAM Journal on Control and Optimization, 34 (1996), 329.  doi: 10.1137/S0363012993247159.  Google Scholar

[2]

I. Bardhan, Consumption and investment under constraints,, Journal of Economic Dynamics and Control, 18 (1994), 909.   Google Scholar

[3]

X. S. Chen and F. H. Yi, A problem of singular stochastic control with optimal stopping in finite horizon,, SIAM Journal on Control and Optimization, 50 (2012), 2151.  doi: 10.1137/110832264.  Google Scholar

[4]

M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations,, Trans. AMS, 277 (1983), 1.  doi: 10.1090/S0002-9947-1983-0690039-8.  Google Scholar

[5]

J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization,, Annals of Applied Probability, 2 (1992), 767.  doi: 10.1214/aoap/1177005576.  Google Scholar

[6]

J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios,, Annals of Applied Probability, 3 (1993), 652.  doi: 10.1214/aoap/1177005357.  Google Scholar

[7]

M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity,, Mathematical Finance, 21 (2011), 775.  doi: 10.1111/j.1467-9965.2010.00449.x.  Google Scholar

[8]

M. Dai, Z. Q. Xu and X. Y. Zhou, Continuous-time mean-variance portfolio selection with proportional transaction costs,, SIAM Journal on Financial Mathematics, 1 (2010), 96.  doi: 10.1137/080742889.  Google Scholar

[9]

M. Dai and F. H. Yi, Finite horizon optimal investment with transaction costs: A parabolic double obstacle problem,, Journal of Differential Equations, 246 (2009), 1445.  doi: 10.1016/j.jde.2008.11.003.  Google Scholar

[10]

M. H. A. Davis and A. Norman, Portfolio selection with transaction costs,, Mathematics of Operations Research, 15 (1990), 676.  doi: 10.1287/moor.15.4.676.  Google Scholar

[11]

R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint,, Finance and Stochastics, 12 (2008), 299.  doi: 10.1007/s00780-008-0066-8.  Google Scholar

[12]

W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Second edition. Stochastic Modelling and Applied Probability, (2006).   Google Scholar

[13]

W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints,, Mathematics of Operations Research, 16 (1991), 802.  doi: 10.1287/moor.16.4.802.  Google Scholar

[14]

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 2,, Communications in Partial Differential Equations, 8 (1983), 1229.  doi: 10.1080/03605308308820301.  Google Scholar

[15]

H. Markowitz, Portfolio selection,, Journal of Finance, 7 (1952), 77.  doi: 10.1111/j.1540-6261.1952.tb01525.x.  Google Scholar

[16]

H. Markowitz, Portfolio Selection: Efficient Diversification of Investments,, John Wiley & Sons, (1959).   Google Scholar

[17]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case,, Review of Economics and Statistics, 51 (1969), 247.  doi: 10.2307/1926560.  Google Scholar

[18]

R. C. Merton, Optimum consumption and portfolio rules in a continuous time model,, Journal of Economic Theory, 3 (1971), 373.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[19]

R. C. Merton, Theory of finance from the perspective of continuous time,, Journal of Financial and Quantitative Analysis, 10 (1975), 659.  doi: 10.2307/2330617.  Google Scholar

[20]

P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming,, Review of Economics and Statistics, 51 (1969), 239.   Google Scholar

[21]

P. S. Sethi, Optimal Consumption and Investment with Bankruptcy,, Kluwer Academic Publishers, (1997).  doi: 10.1007/978-1-4615-6257-3.  Google Scholar

[22]

S. Shreve and M. Soner, Optimal investment and consumption with transaction costs,, Annals of Applied Probability, 4 (1994), 609.  doi: 10.1214/aoap/1177004966.  Google Scholar

[23]

T. Zariphopoulou, Investment-consumption models with transaction fees and Markov chain parameters,, SIAM Journal on Control and Optimization, 30 (1992), 613.  doi: 10.1137/0330035.  Google Scholar

[24]

T. Zariphopoulou, Consumption-investment models with constraints,, SIAM Journal on Control and Optimization, 32 (1994), 59.  doi: 10.1137/S0363012991218827.  Google Scholar

[1]

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
[2]

Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial & Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743
[3]

Qian Zhao, Rongming Wang, Jiaqin Wei. Time-inconsistent consumption-investment problem for a member in a defined contribution pension plan. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1557-1585. doi: 10.3934/jimo.2016.12.1557
[4]

Chonghu Guan, Xun Li, Zuo Quan Xu, Fahuai Yi. A stochastic control problem and related free boundaries in finance. Mathematical Control & Related Fields, 2017, 7 (4) : 563-584. doi: 10.3934/mcrf.2017021
[5]

Lei Sun, Lihong Zhang. Optimal consumption and investment under irrational beliefs. Journal of Industrial & Management Optimization, 2011, 7 (1) : 139-156. doi: 10.3934/jimo.2011.7.139
[6]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019099
[7]

Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315
[8]

Min Dai, Zhou Yang. A note on finite horizon optimal investment and consumption with transaction costs. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1445-1454. doi: 10.3934/dcdsb.2016005
[9]

Yong Ma, Shiping Shan, Weidong Xu. Optimal investment and consumption in the market with jump risk and capital gains tax. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1937-1953. doi: 10.3934/jimo.2018130
[10]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks & Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501
[11]

Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066
[12]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791
[13]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[14]

Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241
[15]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
[16]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020
[17]

Jingshi Li, Jiachuan Zhang, Guoliang Ju, Juntao You. A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations. Electronic Research Archive, 2020, 28 (2) : 977-1000. doi: 10.3934/era.2020052
[18]

C. Burgos, J.-C. Cortés, L. Shaikhet, R.-J. Villanueva. A delayed nonlinear stochastic model for cocaine consumption: Stability analysis and simulation using real data. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020356
[19]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[20]

Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences