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Optimal consumption choice under uncertainty with intertemporal substitution

  • Bank, Peter
  • Riedel, Frank

We extend the analysis of the intertemporal utility maximization problem for Hindy-Huang-Kreps utilities reported in Bank and Riedel (1998) to the stochastic case. Existence and uniqueness of optimal consumption plans are established under arbitrary convex portfolio constraints, including both complete and incomplete markets. For the complete market setting, Kuhn-Tuckerlike necessary and sufficient conditions for optimality are given. Using this characterization, we show that optimal consumption plans are obtained by re- flecting the associated level of satisfaction on a stochastic lower bound. When uncertainty is generated by a Lévy process and agents exhibit constant relative risk aversion, closed-form solutions are derived. Depending on the structure of the underlying stochastics, optimal consumption occurs at rates, in gulps, or singular to Lebesgue measure.

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Paper provided by Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes in its series SFB 373 Discussion Papers with number 1999,71.

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Date of creation: 1999
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Handle: RePEc:zbw:sfb373:199971
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  1. G. Constantinides, 1990. "Habit formation: a resolution of the equity premium puzzle," Levine's Working Paper Archive 1397, David K. Levine.
  2. Cuoco, Domenico, 1997. "Optimal Consumption and Equilibrium Prices with Portfolio Constraints and Stochastic Income," Journal of Economic Theory, Elsevier, vol. 72(1), pages 33-73, January.
  3. Föllmer, Hans & Kramkov, D. O., 1997. "Optional decompositions under constraints," SFB 373 Discussion Papers 1997,31, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  4. Hindy, Ayman & Huang, Chi-fu & Kreps, David, 1992. "On intertemporal preferences in continuous time : The case of certainty," Journal of Mathematical Economics, Elsevier, vol. 21(5), pages 401-440.
  5. H. Föllmer & Y.M. Kabanov, 1997. "Optional decomposition and Lagrange multipliers," Finance and Stochastics, Springer, vol. 2(1), pages 69-81.
  6. Hindy, Ayman & Huang, Chi-fu, 1992. "Intertemporal Preferences for Uncertain Consumption: A Continuous Time Approach," Econometrica, Econometric Society, vol. 60(4), pages 781-801, July.
  7. Bank, Peter & Riedel, Frank, 2000. "Non-time additive utility optimization--the case of certainty," Journal of Mathematical Economics, Elsevier, vol. 33(3), pages 271-290, April.
  8. Sundaresan, Suresh M, 1989. "Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth," Review of Financial Studies, Society for Financial Studies, vol. 2(1), pages 73-89.
  9. Cox, John C. & Huang, Chi-fu, 1989. "Optimal consumption and portfolio policies when asset prices follow a diffusion process," Journal of Economic Theory, Elsevier, vol. 49(1), pages 33-83, October.
  10. D. Vallière & E. Denis & Y. Kabanov, 2009. "Hedging of American options under transaction costs," Finance and Stochastics, Springer, vol. 13(1), pages 105-119, January.
  11. Jin, Xing & Deng, Shuhui, 1997. "Existence and uniqueness of optimal consumption and portfolio rules in a continuous-time finance model with habit formation and without short sales," Journal of Mathematical Economics, Elsevier, vol. 28(2), pages 187-205, September.
  12. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
  13. Duffie, Darrell & Skiadas, Costis, 1994. "Continuous-time security pricing : A utility gradient approach," Journal of Mathematical Economics, Elsevier, vol. 23(2), pages 107-131, March.
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