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October  2013, 9(4): 743-768. doi: 10.3934/jimo.2013.9.743

## 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  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.
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, Cambridge university Press, New York, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar [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.  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), 144-153. 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), 33-83. 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), 465-487. 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), 33-73. 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), 767-818. 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), 381-408. 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), 215-260. 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), 1-10. 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), 259-304. 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), 1557-1586. 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), 203-212. 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), 904-950. 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), 1384-1388.  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), 83-95.  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), 531-547. 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), 247-257. 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), 373-413. 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), 125-136. doi: 10.1080/14697680701213868.  Google Scholar [21] S. R. Pliska, A stochastic calculus model of continuous trading: Optimal portfolio, Math. Oper. Res., 11 (1986), 371-382. 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), 341-360. 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), 1317-1334. 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), 1979-989. doi: 10.1016/j.automatica.2010.02.027.  Google Scholar

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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.  Google Scholar [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.  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), 144-153. 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), 33-83. 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), 465-487. 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), 33-73. 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), 767-818. 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), 381-408. 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), 215-260. 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), 1-10. 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), 259-304. 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), 1557-1586. 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), 203-212. 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), 904-950. 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), 1384-1388.  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), 83-95.  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), 531-547. 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), 247-257. 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), 373-413. 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), 125-136. doi: 10.1080/14697680701213868.  Google Scholar [21] S. R. Pliska, A stochastic calculus model of continuous trading: Optimal portfolio, Math. Oper. Res., 11 (1986), 371-382. 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), 341-360. 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), 1317-1334. 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), 1979-989. doi: 10.1016/j.automatica.2010.02.027.  Google Scholar
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