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A dynamic maximum principle for the optimization of recursive utilities under constraints. (English) Zbl 1040.91038

There is considered the optimization problem when the utility is recursive with constrains on the wealth, which include the case of a large investor or the case of taxes. In other terms, the utility and the wealth processes are supposed to satisfy nonlinear equations. In this work it is shown a backward formulation of this problem which emphasizes the symmetry between utility and wealth. The obtained results can be obtained in the financial engineering domain.

MSC:

91B16 Utility theory
93E20 Optimal stochastic control
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[1] Aubin, J. P. (1984). L’analyse non linéaire et ses motivations économiques. Masson, Paris. · Zbl 0551.90001
[2] Brezis, H. (1983). Analyse fonctionnelle. Masson, Paris. · Zbl 0511.46001
[3] Chen, Z. and Epstein, L. (1999). Ambiguity, risk and asset returns in continuous time. Unpublished manuscript. · Zbl 1121.91359
[4] Constantinides, G. (1982). Intertemporal asset pricingwith heterogenous consumers and without demand aggregation. J.Business 55 253-267.
[5] Constantinides, G. (1986). Capital market equilibrium with transaction costs. J.Political Econom. 94 842-862.
[6] Cox, J. and Huang, C. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J.Econom.Theory 49 33-83. · Zbl 0678.90011 · doi:10.1016/0022-0531(89)90067-7
[7] Cuoco, D. (1997). Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J.Econom.Theory 72 33-73. · Zbl 0883.90050 · doi:10.1006/jeth.1996.2207
[8] Cuoco, D. and Cvitanic, J. (1995). Optimal consumption choices for a ”large” investor. Mimeo, Wharton School, Univ. Pennsylvania. · Zbl 0902.90031 · doi:10.1016/S0165-1889(97)00065-1
[9] Cvitanic, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl.Probab. 2 767-818. · Zbl 0770.90002 · doi:10.1214/aoap/1177005576
[10] Cvitanic, J. and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann. Appl.Probab. 3 652-681. · Zbl 0825.93958 · doi:10.1214/aoap/1177005357
[11] Duffie, D. (1988). Security Markets: Stochastic Models. Academic Press, Boston. · Zbl 0661.90001
[12] Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353-394. JSTOR: · Zbl 0768.90006 · doi:10.1016/0304-4068(92)90028-6
[13] Duffie, D., Fleming, W. and Zariphopoulou, T. (1991). Hedging in incomplete markets with HARA Utility. Research paper 1158, Graduate School of Business, Stanford Univ. · Zbl 0899.90026
[14] Duffie, D. and Skiadas, C. (1994). Continuous-time security pricing: a utility gradient approach. J.Math.Econom.23 107-131. · Zbl 0804.90017 · doi:10.1016/0304-4068(94)90001-9
[15] Duffie, D. and Zariphopoulou, T. (1993). Optimal investment with undiversifiable income risk. Math.Finance 3 135-148. · Zbl 0884.90062 · doi:10.1111/j.1467-9965.1993.tb00083.x
[16] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s and related obstacle problems for PDEs. Ann.Probab.25 702-737. · Zbl 0899.60047 · doi:10.1214/aop/1024404416
[17] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math.Finance 7 1-71. · Zbl 0884.90035 · doi:10.1111/1467-9965.00022
[18] El Karoui, N. and Quenez, M. C. (1995). Dynamic programmingand pricingof contingent claims in incomplete market. SIAM J.Control Optim.33 29-66. · Zbl 0831.90010 · doi:10.1137/S0363012992232579
[19] Epstein, L. and Wang, T. (1994). Intertemporal asset pricingunder Knightian uncertainty, Econometrica 62 283-322. · Zbl 0799.90016 · doi:10.2307/2951614
[20] Epstein, L. and Zin, S. (1989). Substitution, risk aversion and the temporal behavior of consumption and asset returns: a theoritical framework. Econometrica 57 937-969. JSTOR: · Zbl 0683.90012 · doi:10.2307/1913778
[21] Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Control. Springer, New York. · Zbl 0323.49001
[22] Geoffard, P. Y. (1996). Discountingand optimizing: capital accumulation as a variational minmax problem. J.Econom.Theory 69 53-70. · Zbl 0852.90041 · doi:10.1006/jeth.1996.0037
[23] Haussmann, U. (1976). General necessary conditions for optimal control of stochastic system. Math.Program.Study 6 34-38. · Zbl 0369.93048
[24] He, H. and Pearson, N. (1991). Consumption and portfolio policies with incomplete markets and short-sellingconstraints: the infinite-dimensional case. J.Econom.Theory 54 259-304. · Zbl 0736.90017 · doi:10.1016/0022-0531(91)90123-L
[25] Hu, Y. and Peng, S. (1995). Solution of a forward-backward stochastic differential equation. Probab.Theory Related Fields 103, 273-283. · Zbl 0831.60065 · doi:10.1007/BF01204218
[26] Jouini, E. and Kallal, H. (1995). Arbitrage and equilibrium in securities markets with shortsale constraints. Math.Finance 5 197-232. · Zbl 0866.90032 · doi:10.1111/j.1467-9965.1995.tb00065.x
[27] Karatzas, I. (1989). Optimization problems in the theory of continuous trading. SIAM.J.Control Optim. 27 1221-1259. · Zbl 0701.90008 · doi:10.1137/0327063
[28] Karatzas, I., Lehoczky, J. P. and Shreve, S. (1987). Optimal portfolio and consumption decisions for a small investor on a finite horizon. SIAM J.Control Optim.25 1557-1586. · Zbl 0644.93066 · doi:10.1137/0325086
[29] Karatzas, I., Lehoczky, J. P., Shreve, S. and Xu, G. L. (1991). Martingale and duality methods for utility maximisation in an incomplete market, SIAM J.Control Optim.29 702-730. · Zbl 0733.93085 · doi:10.1137/0329039
[30] Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer, Berlin. · Zbl 0638.60065
[31] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann.Appl.Probab.9 904-950. · Zbl 0967.91017 · doi:10.1214/aoap/1029962818
[32] Lehoczky, J., Sethi, S. and Shreve, S. (1983). Optimal consumption and investment policies allowingconsumption constraints and bankruptcy. Math.Oper.Res.8 613-636. JSTOR: · Zbl 0526.90009 · doi:10.1287/moor.8.4.613
[33] Luenberger, D. (1969). Optimization by Vector Space Methods. Wiley, New York. · Zbl 0176.12701
[34] Ma, J., Protter, P. and Yong, J. (1994). Solvingforward-backward stochastic differential equations explicitely-a four step scheme. Probab.Theory Related Fields 98 339-359. · Zbl 0794.60056 · doi:10.1007/BF01192258
[35] Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model. J. Econom.Theory 3 373-413. · Zbl 1011.91502 · doi:10.1016/0022-0531(71)90038-X
[36] Pardoux, P. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Syst.Control Lett.14, 55-61. · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[37] Peng, S. (1992). A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics 38 119-134. · Zbl 0756.49015
[38] Pliska, S. R. (1986). A stochastic calculus model of continuous trading: optimal portfolios. Math. Oper.Res.11 371-382. · Zbl 1011.91503 · doi:10.1287/moor.11.2.371
[39] Schroder, M. and Skiadas, C. (1997). Optimal consumption and portfolio selection with stochastic differential utility. Workingpaper 226, KelloggSchool. · Zbl 0934.91029
[40] Xu, G. L. (1990). A duality approach to a consumption-portfolio decision problem in a continuous market with short-sellingprohibition, Ph.D. disseration, Univ. Pittsburgh.
[41] Xu, G. L. and Shreve, S. (1992). A duality method for optimal consumption and investment under short-sellingprohibition I-general market coefficients; II-constant market coefficients. Ann.Appl.Probab.2 87-112, 314-328. · Zbl 0773.90017 · doi:10.1214/aoap/1177005706
[42] Zariphopoulou, T. (1994). Consumption-investment models with constraints. SIAM J.Control Optim. 32 59-85. · Zbl 0790.90007 · doi:10.1137/S0363012991218827
[43] CMAP, Ecole Polytechnique F-91128 Palaiseau Cedex France E-mail: elkaroui@cmapx.polytechnique.fr S. Peng Institute of Mathematics Shandong University Jinan, 250100 China E-mail: pengsg@shandong.ihep.ac.cn
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