An optimal consumption-investment model with constraint on consumption

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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

Abstract
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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 and 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), 329-364. doi: 10.1137/S0363012993247159.
[2]	I. Bardhan, Consumption and investment under constraints, Journal of Economic Dynamics and Control, 18 (1994), 909-929.
[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), 2151-2172. doi: 10.1137/110832264.
[4]	M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. AMS, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[5]	J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Annals of Applied Probability, 2 (1992), 767-818. doi: 10.1214/aoap/1177005576.
[6]	J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios, Annals of Applied Probability, 3 (1993), 652-681. doi: 10.1214/aoap/1177005357.
[7]	M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity, Mathematical Finance, 21 (2011), 775-793. doi: 10.1111/j.1467-9965.2010.00449.x.
[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), 96-125. doi: 10.1137/080742889.
[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), 1445-1469. doi: 10.1016/j.jde.2008.11.003.
[10]	M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713. doi: 10.1287/moor.15.4.676.
[11]	R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint, Finance and Stochastics, 12 (2008), 299-330. doi: 10.1007/s00780-008-0066-8.
[12]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.
[13]	W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints, Mathematics of Operations Research, 16 (1991), 802-822. doi: 10.1287/moor.16.4.802.
[14]	P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 2, Communications in Partial Differential Equations, 8 (1983), 1229-1276. doi: 10.1080/03605308308820301.
[15]	H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. doi: 10.1111/j.1540-6261.1952.tb01525.x.
[16]	H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, New York, 1959.
[17]	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.
[18]	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.
[19]	R. C. Merton, Theory of finance from the perspective of continuous time, Journal of Financial and Quantitative Analysis, 10 (1975), 659-674. doi: 10.2307/2330617.
[20]	P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239-246.
[21]	P. S. Sethi, Optimal Consumption and Investment with Bankruptcy, Kluwer Academic Publishers, Norwell, MA, 1997. doi: 10.1007/978-1-4615-6257-3.
[22]	S. Shreve and M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609-692. doi: 10.1214/aoap/1177004966.
[23]	T. Zariphopoulou, Investment-consumption models with transaction fees and Markov chain parameters, SIAM Journal on Control and Optimization, 30 (1992), 613-636. doi: 10.1137/0330035.
[24]	T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85. doi: 10.1137/S0363012991218827.

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), 329-364. doi: 10.1137/S0363012993247159.
[2]	I. Bardhan, Consumption and investment under constraints, Journal of Economic Dynamics and Control, 18 (1994), 909-929.
[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), 2151-2172. doi: 10.1137/110832264.
[4]	M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. AMS, 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[5]	J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Annals of Applied Probability, 2 (1992), 767-818. doi: 10.1214/aoap/1177005576.
[6]	J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios, Annals of Applied Probability, 3 (1993), 652-681. doi: 10.1214/aoap/1177005357.
[7]	M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity, Mathematical Finance, 21 (2011), 775-793. doi: 10.1111/j.1467-9965.2010.00449.x.
[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), 96-125. doi: 10.1137/080742889.
[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), 1445-1469. doi: 10.1016/j.jde.2008.11.003.
[10]	M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676-713. doi: 10.1287/moor.15.4.676.
[11]	R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint, Finance and Stochastics, 12 (2008), 299-330. doi: 10.1007/s00780-008-0066-8.
[12]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006.
[13]	W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints, Mathematics of Operations Research, 16 (1991), 802-822. doi: 10.1287/moor.16.4.802.
[14]	P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 2, Communications in Partial Differential Equations, 8 (1983), 1229-1276. doi: 10.1080/03605308308820301.
[15]	H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. doi: 10.1111/j.1540-6261.1952.tb01525.x.
[16]	H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, New York, 1959.
[17]	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.
[18]	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.
[19]	R. C. Merton, Theory of finance from the perspective of continuous time, Journal of Financial and Quantitative Analysis, 10 (1975), 659-674. doi: 10.2307/2330617.
[20]	P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239-246.
[21]	P. S. Sethi, Optimal Consumption and Investment with Bankruptcy, Kluwer Academic Publishers, Norwell, MA, 1997. doi: 10.1007/978-1-4615-6257-3.
[22]	S. Shreve and M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609-692. doi: 10.1214/aoap/1177004966.
[23]	T. Zariphopoulou, Investment-consumption models with transaction fees and Markov chain parameters, SIAM Journal on Control and Optimization, 30 (1992), 613-636. doi: 10.1137/0330035.
[24]	T. Zariphopoulou, Consumption-investment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 59-85. doi: 10.1137/S0363012991218827.

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