American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand
  • JIMO Home
  • This Issue
  • Next Article
    Augmented Lagrange primal-dual approach for generalized fractional programming problems
October  2013, 9(4): 743-768. doi: 10.3934/jimo.2013.9.743

Optimal investment-consumption problem with constraint

Jingzhen Liu 1, , Ka-Fai Cedric Yiu 2, and  Kok Lay Teo 3,

1. 

School of Insurance, Central University Of Finance and Economics, Beijing 100081
2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
3. 

Department of Mathematics and Statistics, Curtin University, Perth, W.A. 6845

Received  September 2012 Revised  April 2013 Published  August 2013

In this paper, we consider an optimal investment-consumption problem subject to a closed convex constraint. In the problem, a constraint is imposed on both the investment and the consumption strategy, rather than just on the investment. The existence of solution is established by using the Martingale technique and convex duality. In addition to investment, our technique embeds also the consumption into a family of fictitious markets. However, with the addition of consumption, it leads to nonreflexive dual spaces. This difficulty is overcome by employing the so-called technique of ``relaxation-projection" to establish the existence of solution to the problem. Furthermore, if the solution to the dual problem is obtained, then the solution to the primal problem can be found by using the characterization of the solution. An illustrative example is given with a dynamic risk constraint to demonstrate the method.
Keywords: Investment, martingale, consumption, dynamic risk constraint., duality.
Mathematics Subject Classification: Primary: 60G46, 60H99; Secondary: 90C1.
Citation: Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial and Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743
References:
[1]

D. Applebeaum, "Levy Processes and Stochastic Calculus," $2^{nd}$ edition, Cambridge university Press, New York, 2009. doi: 10.1017/CBO9780511809781.

[2]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, Math. Anal. Appl., 44 (1974), 384-404. doi: 10.1016/0022-247X(73)90066-8.

[3]

S. M. Chen, Z. F. Li and K. M. Li, Optimal investment-einsurance policy for an insurance company with VaR constraint, Insurance: Mathematics and Economics, 47 (2010), 144-153. doi: 10.1016/j.insmatheco.2010.06.002.

[4]

J. C. Cox and C. F. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econom. Theory., 49 (1989), 33-83. doi: 10.1016/0022-0531(89)90067-7.

[5]

J. C. Cox and C. F. Huang, A variational problem arising in financial economics, J. Math. Econom., 20 (1991), 465-487. doi: 10.1016/0304-4068(91)90004-D.

[6]

D. Cuoco, Optimal consumption and equilibrium prices with portfolio constraints and stochastic income, J. Econom. Theory., 72 (1997), 33-73. doi: 10.1006/jeth.1996.2207.

[7]

J. Cvitanic and I. Karatzas, Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2 (1992), 767-818. doi: 10.1214/aoap/1177005576.

[8]

J. M. Harrison and D. Kreps, Martingales and arbitrage in multiperiod security markets, J. Econom. Theory., 20 (1979), 381-408. doi: 10.1016/0022-0531(79)90043-7.

[9]

J. M. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl., 11 (1981), 215-260. doi: 10.1016/0304-4149(81)90026-0.

[10]

H. He and N. D. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The finite-dimensional case, Mathematical Finance, 1 (1991), 1-10. doi: 10.1016/0022-0531(91)90123-L.

[11]

H. He and N. D. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-dimensional case, J. Econom. Theory., 54 (1991), 259-304. doi: 10.1016/0022-0531(91)90123-L.

[12]

I. Karatzas, J. P. Lehoczky and S. E. Shreve, Optimal portfolio and consumption decisions for a small investor on a finite horizon, SIAM J. Control Optim., 25 (1987), 1557-1586. doi: 10.1137/0325086.

[13]

I. Karatzas, J. P. Lehoczky, S. E. Shreve and G. L. Xu, Martingale and duality methods for utility maximization in incomplete markets, Mathematical Finance, 15 (1991), 203-212. doi: 10.1137/0329039.

[14]

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab., 9 (1999), 904-950. doi: 10.1214/aoap/1029962818.

[15]

V. L. Levin, Extreme problems with convex functionals that are lower-semicontinuous with respect to convergence in measure, Soviet math. Dokl., 16 (1976), 1384-1388.

[16]

J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal portfolios with stress analysis and the effect of a CVaR constraint, Pac. J. Optim., 7 (2011), 83-95.

[17]

J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547. doi: 10.3934/jimo.2012.8.531.

[18]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560.

[19]

R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, J. Econom. Theory., 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.

[20]

T. A. Pirvu, Portfolio optimization under the Value-at-Risk constraint, Quantitative Finance, 7 (2007), 125-136. doi: 10.1080/14697680701213868.

[21]

S. R. Pliska, A stochastic calculus model of continuous trading: Optimal portfolio, Math. Oper. Res., 11 (1986), 371-382. doi: 10.1287/moor.11.2.371.

[22]

S. A. Ross, The arbitrage theory of capital asset pricing, J. Econom. Theory., 13 (1976), 341-360. doi: 10.1016/0022-0531(76)90046-6.

[23]

K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint, Journal of Economic Dynamics and Control, 28 (2004), 1317-1334. doi: 10.1016/S0165-1889(03)00116-7.

[24]

K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 1979-989. doi: 10.1016/j.automatica.2010.02.027.

show all references

References:
[1]

D. Applebeaum, "Levy Processes and Stochastic Calculus," $2^{nd}$ edition, Cambridge university Press, New York, 2009. doi: 10.1017/CBO9780511809781.

[2]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, Math. Anal. Appl., 44 (1974), 384-404. doi: 10.1016/0022-247X(73)90066-8.

[3]

S. M. Chen, Z. F. Li and K. M. Li, Optimal investment-einsurance policy for an insurance company with VaR constraint, Insurance: Mathematics and Economics, 47 (2010), 144-153. doi: 10.1016/j.insmatheco.2010.06.002.

[4]

J. C. Cox and C. F. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econom. Theory., 49 (1989), 33-83. doi: 10.1016/0022-0531(89)90067-7.

[5]

J. C. Cox and C. F. Huang, A variational problem arising in financial economics, J. Math. Econom., 20 (1991), 465-487. doi: 10.1016/0304-4068(91)90004-D.

[6]

D. Cuoco, Optimal consumption and equilibrium prices with portfolio constraints and stochastic income, J. Econom. Theory., 72 (1997), 33-73. doi: 10.1006/jeth.1996.2207.

[7]

J. Cvitanic and I. Karatzas, Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2 (1992), 767-818. doi: 10.1214/aoap/1177005576.

[8]

J. M. Harrison and D. Kreps, Martingales and arbitrage in multiperiod security markets, J. Econom. Theory., 20 (1979), 381-408. doi: 10.1016/0022-0531(79)90043-7.

[9]

J. M. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl., 11 (1981), 215-260. doi: 10.1016/0304-4149(81)90026-0.

[10]

H. He and N. D. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The finite-dimensional case, Mathematical Finance, 1 (1991), 1-10. doi: 10.1016/0022-0531(91)90123-L.

[11]

H. He and N. D. Pearson, Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-dimensional case, J. Econom. Theory., 54 (1991), 259-304. doi: 10.1016/0022-0531(91)90123-L.

[12]

I. Karatzas, J. P. Lehoczky and S. E. Shreve, Optimal portfolio and consumption decisions for a small investor on a finite horizon, SIAM J. Control Optim., 25 (1987), 1557-1586. doi: 10.1137/0325086.

[13]

I. Karatzas, J. P. Lehoczky, S. E. Shreve and G. L. Xu, Martingale and duality methods for utility maximization in incomplete markets, Mathematical Finance, 15 (1991), 203-212. doi: 10.1137/0329039.

[14]

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab., 9 (1999), 904-950. doi: 10.1214/aoap/1029962818.

[15]

V. L. Levin, Extreme problems with convex functionals that are lower-semicontinuous with respect to convergence in measure, Soviet math. Dokl., 16 (1976), 1384-1388.

[16]

J. Z. Liu, K. F. C. Yiu and K. L. Teo, Optimal portfolios with stress analysis and the effect of a CVaR constraint, Pac. J. Optim., 7 (2011), 83-95.

[17]

J. Z. Liu, L. H. Bai and K. F. C. Yiu, Optimal investment with a value-at-risk constraint, Journal of Industrial and Management Optimization, 8 (2012), 531-547. doi: 10.3934/jimo.2012.8.531.

[18]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257. doi: 10.2307/1926560.

[19]

R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, J. Econom. Theory., 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X.

[20]

T. A. Pirvu, Portfolio optimization under the Value-at-Risk constraint, Quantitative Finance, 7 (2007), 125-136. doi: 10.1080/14697680701213868.

[21]

S. R. Pliska, A stochastic calculus model of continuous trading: Optimal portfolio, Math. Oper. Res., 11 (1986), 371-382. doi: 10.1287/moor.11.2.371.

[22]

S. A. Ross, The arbitrage theory of capital asset pricing, J. Econom. Theory., 13 (1976), 341-360. doi: 10.1016/0022-0531(76)90046-6.

[23]

K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint, Journal of Economic Dynamics and Control, 28 (2004), 1317-1334. doi: 10.1016/S0165-1889(03)00116-7.

[24]

K. F. C. Yiu, J. Z. Liu, T. K. Siu and W. C. Ching, Optimal portfolios with regime-switching and value-at-risk constraint, Automatica, 46 (2010), 1979-989. doi: 10.1016/j.automatica.2010.02.027.

[1]

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
[2]

Jingzhen Liu, Lihua Bai, Ka-Fai Cedric Yiu. Optimal investment with a value-at-risk constraint. Journal of Industrial and Management Optimization, 2012, 8 (3) : 531-547. doi: 10.3934/jimo.2012.8.531
[3]

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

Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2195-2211. doi: 10.3934/jimo.2019050
[5]

Hassen Aydi, Ayman Kachmar. Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II. Communications on Pure and Applied Analysis, 2009, 8 (3) : 977-998. doi: 10.3934/cpaa.2009.8.977
[6]

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

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
[8]

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

Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1915-1934. doi: 10.3934/jimo.2021049
[10]

Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315
[11]

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

Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control and Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017
[13]

Jingzhen Liu, Shiqi Yan, Shan Jiang, Jiaqin Wei. Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022040
[14]

Chao Deng, Haixiang Yao, Yan Chen. Optimal investment and risk control problems with delay for an insurer in defaultable market. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2563-2579. doi: 10.3934/jimo.2019070
[15]

K. F. Cedric Yiu, S. Y. Wang, K. L. Mak. Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains. Journal of Industrial and Management Optimization, 2008, 4 (1) : 81-94. doi: 10.3934/jimo.2008.4.81
[16]

Vladimir Korotkov, Vladimir Emelichev, Yury Nikulin. Multicriteria investment problem with Savage's risk criteria: Theoretical aspects of stability and case study. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1297-1310. doi: 10.3934/jimo.2019003
[17]

Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1145-1160. doi: 10.3934/jimo.2021013
[18]

Tomasz R. Bielecki, Igor Cialenco, Marcin Pitera. A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 3-. doi: 10.1186/s41546-017-0012-9
[19]

Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial and Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010
[20]

Rongfei Liu, Dingcheng Wang, Jiangyan Peng. Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial and Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058

2021 Impact Factor: 1.411

Tools

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy