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 & 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, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[2]

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

[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.  doi: 10.1016/j.insmatheco.2010.06.002.  Google Scholar

[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.  doi: 10.1016/0022-0531(89)90067-7.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1016/0022-0531(91)90123-L.  Google Scholar

[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.  doi: 10.1016/0022-0531(91)90123-L.  Google Scholar

[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.  doi: 10.1137/0325086.  Google Scholar

[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.  doi: 10.1137/0329039.  Google Scholar

[14]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2012.8.531.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

show all references

References:
[1]

D. Applebeaum, "Levy Processes and Stochastic Calculus,", $2^{nd}$ edition, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[2]

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

[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.  doi: 10.1016/j.insmatheco.2010.06.002.  Google Scholar

[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.  doi: 10.1016/0022-0531(89)90067-7.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  doi: 10.1016/0022-0531(91)90123-L.  Google Scholar

[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.  doi: 10.1016/0022-0531(91)90123-L.  Google Scholar

[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.  doi: 10.1137/0325086.  Google Scholar

[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.  doi: 10.1137/0329039.  Google Scholar

[14]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2012.8.531.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

[1]

Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021013
[2]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166
[3]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285
[4]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176
[5]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071
[6]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391
[7]

Haili Yuan, Yijun Hu. Optimal investment for an insurer under liquid reserves. Journal of Industrial & Management Optimization, 2021, 17 (1) : 339-355. doi: 10.3934/jimo.2019114
[8]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072
[9]

Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226
[10]

Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021002
[11]

Ripeng Huang, Shaojian Qu, Xiaoguang Yang, Zhimin Liu. Multi-stage distributionally robust optimization with risk aversion. Journal of Industrial & Management Optimization, 2021, 17 (1) : 233-259. doi: 10.3934/jimo.2019109
[12]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100
[13]

Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117
[14]

Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021003
[15]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127
[16]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008
[17]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051
[18]

Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134
[19]

P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178
[20]

Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences