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A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations
An optimal consumptioninvestment model with constraint on consumption
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 
References:
[1] 
M. Akian, J. L. Menaldi and A. Sulem, On an investmentconsumption model with transaction costs,, SIAM Journal on Control and Optimization, 34 (1996), 329. doi: 10.1137/S0363012993247159. Google Scholar 
[2] 
I. Bardhan, Consumption and investment under constraints,, Journal of Economic Dynamics and Control, 18 (1994), 909. Google Scholar 
[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. doi: 10.1137/110832264. Google Scholar 
[4] 
M. G. Crandall and P. L. Lions, Viscosity solutions of HamiltonJacobi equations,, Trans. AMS, 277 (1983), 1. doi: 10.1090/S00029947198306900398. Google Scholar 
[5] 
J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization,, Annals of Applied Probability, 2 (1992), 767. doi: 10.1214/aoap/1177005576. Google Scholar 
[6] 
J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios,, Annals of Applied Probability, 3 (1993), 652. doi: 10.1214/aoap/1177005357. Google Scholar 
[7] 
M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity,, Mathematical Finance, 21 (2011), 775. doi: 10.1111/j.14679965.2010.00449.x. Google Scholar 
[8] 
M. Dai, Z. Q. Xu and X. Y. Zhou, Continuoustime meanvariance portfolio selection with proportional transaction costs,, SIAM Journal on Financial Mathematics, 1 (2010), 96. doi: 10.1137/080742889. Google Scholar 
[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. doi: 10.1016/j.jde.2008.11.003. Google Scholar 
[10] 
M. H. A. Davis and A. Norman, Portfolio selection with transaction costs,, Mathematics of Operations Research, 15 (1990), 676. doi: 10.1287/moor.15.4.676. Google Scholar 
[11] 
R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint,, Finance and Stochastics, 12 (2008), 299. doi: 10.1007/s0078000800668. Google Scholar 
[12] 
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Second edition. Stochastic Modelling and Applied Probability, (2006). Google Scholar 
[13] 
W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints,, Mathematics of Operations Research, 16 (1991), 802. doi: 10.1287/moor.16.4.802. Google Scholar 
[14] 
P. L. Lions, Optimal control of diffusion processes and HamiltonJacobiBellman equations, Part 2,, Communications in Partial Differential Equations, 8 (1983), 1229. doi: 10.1080/03605308308820301. Google Scholar 
[15] 
H. Markowitz, Portfolio selection,, Journal of Finance, 7 (1952), 77. doi: 10.1111/j.15406261.1952.tb01525.x. Google Scholar 
[16] 
H. Markowitz, Portfolio Selection: Efficient Diversification of Investments,, John Wiley & Sons, (1959). Google Scholar 
[17] 
R. C. Merton, Lifetime portfolio selection under uncertainty: The continuoustime case,, Review of Economics and Statistics, 51 (1969), 247. doi: 10.2307/1926560. Google Scholar 
[18] 
R. C. Merton, Optimum consumption and portfolio rules in a continuous time model,, Journal of Economic Theory, 3 (1971), 373. doi: 10.1016/00220531(71)90038X. Google Scholar 
[19] 
R. C. Merton, Theory of finance from the perspective of continuous time,, Journal of Financial and Quantitative Analysis, 10 (1975), 659. doi: 10.2307/2330617. Google Scholar 
[20] 
P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming,, Review of Economics and Statistics, 51 (1969), 239. Google Scholar 
[21] 
P. S. Sethi, Optimal Consumption and Investment with Bankruptcy,, Kluwer Academic Publishers, (1997). doi: 10.1007/9781461562573. Google Scholar 
[22] 
S. Shreve and M. Soner, Optimal investment and consumption with transaction costs,, Annals of Applied Probability, 4 (1994), 609. doi: 10.1214/aoap/1177004966. Google Scholar 
[23] 
T. Zariphopoulou, Investmentconsumption models with transaction fees and Markov chain parameters,, SIAM Journal on Control and Optimization, 30 (1992), 613. doi: 10.1137/0330035. Google Scholar 
[24] 
T. Zariphopoulou, Consumptioninvestment models with constraints,, SIAM Journal on Control and Optimization, 32 (1994), 59. doi: 10.1137/S0363012991218827. Google Scholar 
show all references
References:
[1] 
M. Akian, J. L. Menaldi and A. Sulem, On an investmentconsumption model with transaction costs,, SIAM Journal on Control and Optimization, 34 (1996), 329. doi: 10.1137/S0363012993247159. Google Scholar 
[2] 
I. Bardhan, Consumption and investment under constraints,, Journal of Economic Dynamics and Control, 18 (1994), 909. Google Scholar 
[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. doi: 10.1137/110832264. Google Scholar 
[4] 
M. G. Crandall and P. L. Lions, Viscosity solutions of HamiltonJacobi equations,, Trans. AMS, 277 (1983), 1. doi: 10.1090/S00029947198306900398. Google Scholar 
[5] 
J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization,, Annals of Applied Probability, 2 (1992), 767. doi: 10.1214/aoap/1177005576. Google Scholar 
[6] 
J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios,, Annals of Applied Probability, 3 (1993), 652. doi: 10.1214/aoap/1177005357. Google Scholar 
[7] 
M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity,, Mathematical Finance, 21 (2011), 775. doi: 10.1111/j.14679965.2010.00449.x. Google Scholar 
[8] 
M. Dai, Z. Q. Xu and X. Y. Zhou, Continuoustime meanvariance portfolio selection with proportional transaction costs,, SIAM Journal on Financial Mathematics, 1 (2010), 96. doi: 10.1137/080742889. Google Scholar 
[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. doi: 10.1016/j.jde.2008.11.003. Google Scholar 
[10] 
M. H. A. Davis and A. Norman, Portfolio selection with transaction costs,, Mathematics of Operations Research, 15 (1990), 676. doi: 10.1287/moor.15.4.676. Google Scholar 
[11] 
R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint,, Finance and Stochastics, 12 (2008), 299. doi: 10.1007/s0078000800668. Google Scholar 
[12] 
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions,, Second edition. Stochastic Modelling and Applied Probability, (2006). Google Scholar 
[13] 
W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints,, Mathematics of Operations Research, 16 (1991), 802. doi: 10.1287/moor.16.4.802. Google Scholar 
[14] 
P. L. Lions, Optimal control of diffusion processes and HamiltonJacobiBellman equations, Part 2,, Communications in Partial Differential Equations, 8 (1983), 1229. doi: 10.1080/03605308308820301. Google Scholar 
[15] 
H. Markowitz, Portfolio selection,, Journal of Finance, 7 (1952), 77. doi: 10.1111/j.15406261.1952.tb01525.x. Google Scholar 
[16] 
H. Markowitz, Portfolio Selection: Efficient Diversification of Investments,, John Wiley & Sons, (1959). Google Scholar 
[17] 
R. C. Merton, Lifetime portfolio selection under uncertainty: The continuoustime case,, Review of Economics and Statistics, 51 (1969), 247. doi: 10.2307/1926560. Google Scholar 
[18] 
R. C. Merton, Optimum consumption and portfolio rules in a continuous time model,, Journal of Economic Theory, 3 (1971), 373. doi: 10.1016/00220531(71)90038X. Google Scholar 
[19] 
R. C. Merton, Theory of finance from the perspective of continuous time,, Journal of Financial and Quantitative Analysis, 10 (1975), 659. doi: 10.2307/2330617. Google Scholar 
[20] 
P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming,, Review of Economics and Statistics, 51 (1969), 239. Google Scholar 
[21] 
P. S. Sethi, Optimal Consumption and Investment with Bankruptcy,, Kluwer Academic Publishers, (1997). doi: 10.1007/9781461562573. Google Scholar 
[22] 
S. Shreve and M. Soner, Optimal investment and consumption with transaction costs,, Annals of Applied Probability, 4 (1994), 609. doi: 10.1214/aoap/1177004966. Google Scholar 
[23] 
T. Zariphopoulou, Investmentconsumption models with transaction fees and Markov chain parameters,, SIAM Journal on Control and Optimization, 30 (1992), 613. doi: 10.1137/0330035. Google Scholar 
[24] 
T. Zariphopoulou, Consumptioninvestment models with constraints,, SIAM Journal on Control and Optimization, 32 (1994), 59. doi: 10.1137/S0363012991218827. Google Scholar 
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