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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), 329364. doi: 10.1137/S0363012993247159. 
[2] 
I. Bardhan, Consumption and investment under constraints, Journal of Economic Dynamics and Control, 18 (1994), 909929. 
[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), 21512172. doi: 10.1137/110832264. 
[4] 
M. G. Crandall and P. L. Lions, Viscosity solutions of HamiltonJacobi equations, Trans. AMS, 277 (1983), 142. doi: 10.1090/S00029947198306900398. 
[5] 
J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Annals of Applied Probability, 2 (1992), 767818. doi: 10.1214/aoap/1177005576. 
[6] 
J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios, Annals of Applied Probability, 3 (1993), 652681. doi: 10.1214/aoap/1177005357. 
[7] 
M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity, Mathematical Finance, 21 (2011), 775793. doi: 10.1111/j.14679965.2010.00449.x. 
[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), 96125. doi: 10.1137/080742889. 
[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), 14451469. doi: 10.1016/j.jde.2008.11.003. 
[10] 
M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676713. doi: 10.1287/moor.15.4.676. 
[11] 
R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint, Finance and Stochastics, 12 (2008), 299330. doi: 10.1007/s0078000800668. 
[12] 
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. 
[13] 
W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints, Mathematics of Operations Research, 16 (1991), 802822. doi: 10.1287/moor.16.4.802. 
[14] 
P. L. Lions, Optimal control of diffusion processes and HamiltonJacobiBellman equations, Part 2, Communications in Partial Differential Equations, 8 (1983), 12291276. doi: 10.1080/03605308308820301. 
[15] 
H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 7791. doi: 10.1111/j.15406261.1952.tb01525.x. 
[16] 
H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, New York, 1959. 
[17] 
R. C. Merton, Lifetime portfolio selection under uncertainty: The continuoustime case, Review of Economics and Statistics, 51 (1969), 247257. doi: 10.2307/1926560. 
[18] 
R. C. Merton, Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3 (1971), 373413. doi: 10.1016/00220531(71)90038X. 
[19] 
R. C. Merton, Theory of finance from the perspective of continuous time, Journal of Financial and Quantitative Analysis, 10 (1975), 659674. doi: 10.2307/2330617. 
[20] 
P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239246. 
[21] 
P. S. Sethi, Optimal Consumption and Investment with Bankruptcy, Kluwer Academic Publishers, Norwell, MA, 1997. doi: 10.1007/9781461562573. 
[22] 
S. Shreve and M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609692. doi: 10.1214/aoap/1177004966. 
[23] 
T. Zariphopoulou, Investmentconsumption models with transaction fees and Markov chain parameters, SIAM Journal on Control and Optimization, 30 (1992), 613636. doi: 10.1137/0330035. 
[24] 
T. Zariphopoulou, Consumptioninvestment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 5985. doi: 10.1137/S0363012991218827. 
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), 329364. doi: 10.1137/S0363012993247159. 
[2] 
I. Bardhan, Consumption and investment under constraints, Journal of Economic Dynamics and Control, 18 (1994), 909929. 
[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), 21512172. doi: 10.1137/110832264. 
[4] 
M. G. Crandall and P. L. Lions, Viscosity solutions of HamiltonJacobi equations, Trans. AMS, 277 (1983), 142. doi: 10.1090/S00029947198306900398. 
[5] 
J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Annals of Applied Probability, 2 (1992), 767818. doi: 10.1214/aoap/1177005576. 
[6] 
J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios, Annals of Applied Probability, 3 (1993), 652681. doi: 10.1214/aoap/1177005357. 
[7] 
M. Dai and Z. Xu, Optimal redeeming strategy of stock loans with finite maturity, Mathematical Finance, 21 (2011), 775793. doi: 10.1111/j.14679965.2010.00449.x. 
[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), 96125. doi: 10.1137/080742889. 
[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), 14451469. doi: 10.1016/j.jde.2008.11.003. 
[10] 
M. H. A. Davis and A. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1990), 676713. doi: 10.1287/moor.15.4.676. 
[11] 
R. Elie and N. Touzi, Optimal lifetime consumption and investment under a drawdown constraint, Finance and Stochastics, 12 (2008), 299330. doi: 10.1007/s0078000800668. 
[12] 
W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. 
[13] 
W. H. Fleming and T. Zariphopoulou, An optimal consumption and investment models with borrowing constraints, Mathematics of Operations Research, 16 (1991), 802822. doi: 10.1287/moor.16.4.802. 
[14] 
P. L. Lions, Optimal control of diffusion processes and HamiltonJacobiBellman equations, Part 2, Communications in Partial Differential Equations, 8 (1983), 12291276. doi: 10.1080/03605308308820301. 
[15] 
H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 7791. doi: 10.1111/j.15406261.1952.tb01525.x. 
[16] 
H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, New York, 1959. 
[17] 
R. C. Merton, Lifetime portfolio selection under uncertainty: The continuoustime case, Review of Economics and Statistics, 51 (1969), 247257. doi: 10.2307/1926560. 
[18] 
R. C. Merton, Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3 (1971), 373413. doi: 10.1016/00220531(71)90038X. 
[19] 
R. C. Merton, Theory of finance from the perspective of continuous time, Journal of Financial and Quantitative Analysis, 10 (1975), 659674. doi: 10.2307/2330617. 
[20] 
P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239246. 
[21] 
P. S. Sethi, Optimal Consumption and Investment with Bankruptcy, Kluwer Academic Publishers, Norwell, MA, 1997. doi: 10.1007/9781461562573. 
[22] 
S. Shreve and M. Soner, Optimal investment and consumption with transaction costs, Annals of Applied Probability, 4 (1994), 609692. doi: 10.1214/aoap/1177004966. 
[23] 
T. Zariphopoulou, Investmentconsumption models with transaction fees and Markov chain parameters, SIAM Journal on Control and Optimization, 30 (1992), 613636. doi: 10.1137/0330035. 
[24] 
T. Zariphopoulou, Consumptioninvestment models with constraints, SIAM Journal on Control and Optimization, 32 (1994), 5985. doi: 10.1137/S0363012991218827. 
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