Optimal investment-consumption    problem with constraint

Jingzhen Liu; Ka-Fai Cedric Yiu; Kok Lay Teo

doi:10.3934/jimo.2013.9.743

Article Contents

2013, Volume 9, Issue 4: 743-768. Doi: 10.3934/jimo.2013.9.743

This issue Previous Article Augmented Lagrange primal-dual approach for generalized fractional programming problems Next Article Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand

Optimal investment-consumption problem with constraint

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: August 31, 2012

Revised: March 31, 2013

Published: September 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 60G46, 60H99; Secondary: 90C15.

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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.