The Annals of Applied Probability

A dynamic maximum principle for the optimization of recursive utilities under constraints

N. El Karoui, S. Peng, and M. C. Quenez

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This paper examines the continuous-time portfolio-consumption problem of an agent with a recursive utility in the presence of nonlinear constraints on the wealth.Using backward stochastic differential equations, we state a dynamic maximum principle which generalizes the characterization of optimal policies obtained by Duffie and Skiadas [J.Math Econ. 23, 107 –131 (1994)] in the case of a linear wealth. From this property, we derive a characterization of optimal wealth and utility processes as the unique solution of a forward-backward system. Existence of an optimal policy is also established via a penalization method.

Article information

Ann. Appl. Probab., Volume 11, Number 3 (2001), 664-693.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 92E20: Classical flows, reactions, etc. [See also 80A30, 80A32]
Secondary: 60J60: Diffusion processes [See also 58J65] 35B50: Maximum principles

Utility maximization recursive utility large investor backward stochastic differential equations maximum principle forward-backward system


El Karoui, N.; Peng, S.; Quenez, M. C. A dynamic maximum principle for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11 (2001), no. 3, 664--693. doi:10.1214/aoap/1015345345.

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  • Aubin, J. P. (1984). L'analyse non lin´eaire et ses motivations ´economiques. Masson, Paris.
  • Brezis, H. (1983). Analyse fonctionnelle. Masson, Paris.
  • Chen, Z. and Epstein, L. (1999). Ambiguity, risk and asset returns in continuous time. Unpublished manuscript.
  • Constantinides, G. (1982). Intertemporal asset pricingwith heterogenous consumers and without demand aggregation. J.Business 55 253-267.
  • Constantinides, G. (1986). Capital market equilibrium with transaction costs. J.Political Econom. 94 842-862.
  • Cox, J. and Huang, C. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J.Econom.Theory 49 33-83.
  • Cuoco, D. (1997). Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J.Econom.Theory 72 33-73.
  • Cuoco, D. and Cvitanic, J. (1995). Optimal consumption choices for a "large" investor. Mimeo, Wharton School, Univ. Pennsylvania.
  • Cvitanic, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl.Probab. 2 767-818.
  • Cvitanic, J. and Karatzas, I. (1993). Hedging contingent claims with constrained portfolios. Ann. Appl.Probab. 3 652-681.
  • Duffie, D. (1988). Security Markets: Stochastic Models. Academic Press, Boston.
  • Duffie, D. and Epstein, L. (1992). Stochastic differential utility. Econometrica 60 353-394.
  • Duffie, D., Fleming, W. and Zariphopoulou, T. (1991). Hedging in incomplete markets with HARA Utility. Research paper 1158, Graduate School of Business, Stanford Univ.
  • Duffie, D. and Skiadas, C. (1994). Continuous-time security pricing: a utility gradient approach. J.Math.Econom.23 107-131.
  • Duffie, D. and Zariphopoulou, T. (1993). Optimal investment with undiversifiable income risk. Math.Finance 3 135-148.
  • 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.
  • El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math.Finance 7 1-71.
  • El Karoui, N. and Quenez, M. C. (1995). Dynamic programmingand pricingof contingent claims in incomplete market. SIAM J.Control Optim.33 29-66.
  • Epstein, L. and Wang, T. (1994). Intertemporal asset pricingunder Knightian uncertainty, Econometrica 62 283-322.
  • 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.
  • Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Control. Springer, New York.
  • Geoffard, P. Y. (1996). Discountingand optimizing: capital accumulation as a variational minmax problem. J.Econom.Theory 69 53-70.
  • Haussmann, U. (1976). General necessary conditions for optimal control of stochastic system. Math.Program.Study 6 34-38.
  • 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.
  • Hu, Y. and Peng, S. (1995). Solution of a forward-backward stochastic differential equation. Probab.Theory Related Fields 103, 273-283.
  • Jouini, E. and Kallal, H. (1995). Arbitrage and equilibrium in securities markets with shortsale constraints. Math.Finance 5 197-232.
  • Karatzas, I. (1989). Optimization problems in the theory of continuous trading. SIAM.J.Control Optim. 27 1221-1259.
  • 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.
  • 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.
  • Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer, Berlin.
  • Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann.Appl.Probab.9 904-950.
  • Lehoczky, J., Sethi, S. and Shreve, S. (1983). Optimal consumption and investment policies allowingconsumption constraints and bankruptcy. Math.Oper.Res.8 613-636.
  • Luenberger, D. (1969). Optimization by Vector Space Methods. Wiley, New York.
  • 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.
  • Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model. J. Econom.Theory 3 373-413.
  • Pardoux, P. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Syst.Control Lett.14, 55-61.
  • Peng, S. (1992). A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics 38 119-134.
  • Pliska, S. R. (1986). A stochastic calculus model of continuous trading: optimal portfolios. Math. Oper.Res.11 371-382.
  • Schroder, M. and Skiadas, C. (1997). Optimal consumption and portfolio selection with stochastic differential utility. Workingpaper 226, KelloggSchool.
  • 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.
  • 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.
  • Zariphopoulou, T. (1994). Consumption-investment models with constraints. SIAM J.Control Optim. 32 59-85.
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