Mathematics and Financial Economics

, Volume 3, Issue 3–4, pp 139–159 | Cite as

The opportunity process for optimal consumption and investment with power utility

Article

Abstract

We study the utility maximization problem for power utility random fields in a semimartingale financial market, with and without intermediate consumption. The notion of an opportunity process is introduced as a reduced form of the value process of the resulting stochastic control problem. We show how the opportunity process describes the key objects: optimal strategy, value function, and dual problem. The results are applied to obtain monotonicity properties of the optimal consumption.

Keywords

Power utility Consumption Semimartingale Dynamic programming Convex duality 

JEL Classification

G11 C61 

Mathematics Subject Classification (2000)

91B28 91B42 93E20 60G44 

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References

  1. 1.
    Černý A., Kallsen J.: On the structure of general mean-variance hedging strategies. Ann. Probab. 35(4), 1479–1531 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Delbaen F., Monat P., Schachermayer W., Schweizer M., Stricker C.: Weighted norm inequalities and hedging in incomplete markets. Finance Stoch. 1(3), 181–227 (1997)MATHCrossRefGoogle Scholar
  3. 3.
    Delbaen F., Schachermayer W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dellacherie C., Meyer P.A.: Probabilities and Potential B. Amsterdam, North Holland (1982)MATHGoogle Scholar
  5. 5.
    Doléans-Dade, C., Meyer, P.A.: Inégalités de normes avec poids. In: Séminaire de Probabilités XIII (1977/78) Lecture Notes in Math., vol. 721, pp. 313–331. Springer, Berlin (1979)Google Scholar
  6. 6.
    El Karoui N., Quenez M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33(1), 29–66 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Goll T., Kallsen J.: A complete explicit solution to the log-optimal portfolio problem. Ann. Appl. Probab. 13(2), 774–799 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gollier C.: The Economics of Risk and Time. MIT Press, Cambridge (2001)MATHGoogle Scholar
  9. 9.
    Jacod J., Shiryaev A.N.: Limit Theorems for Stochastic Processes. 2nd edn. Springer, Berlin (2003)MATHGoogle Scholar
  10. 10.
    Kallsen, J., Muhle-Karbe, J.: Utility maximization in affine stochastic volatility models. Int. J. Theor. Appl. Finance (2008, to appear)Google Scholar
  11. 11.
    Karatzas I., Shreve S.E.: Methods of Mathematical Finance. Springer, New York (1998)MATHGoogle Scholar
  12. 12.
    Karatzas I., Žitković G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31(4), 1821–1858 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kim T.S., Omberg E.: Dynamic nonmyopic portfolio behavior. Rev. Financ. Stud. 9(1), 141–161 (1996)CrossRefGoogle Scholar
  14. 14.
    Kraft H.: Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility. Quant. Finance 5(3), 303–313 (2005)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kramkov D., Schachermayer W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(3), 904–950 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kramkov D., Schachermayer W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13(4), 1504–1516 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Laurent J.P., Pham H.: Dynamic programming and mean-variance hedging. Finance Stoch. 3(1), 83–110 (1999)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Mania M., Tevzadze R.: A unified characterization of the q-optimal and minimal entropy martingale measures. Georgian Math. J. 10(2), 289–310 (2003)MATHMathSciNetGoogle Scholar
  19. 19.
    Merton R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413 (1971)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Muhle-Karbe, J.: On Utility-Based Investment, Pricing and Hedging in Incomplete Markets. PhD thesis, TU München (2009)Google Scholar
  21. 21.
    Nutz, M.: The Bellman equation for power utility maximization with semimartingales. Preprint (arXiv:0912.1883v1) (2009)Google Scholar
  22. 22.
    Nutz, M.: Power utility maximization in constrained exponential L évy models. Preprint (arXiv: 0912.1885v1) (2009)Google Scholar
  23. 23.
    Nutz, M.: Risk aversion asymptotics for power utility maximization. Preprint (arXiv:1003. 3582v1) (2010)Google Scholar
  24. 24.
    Schweizer M.: On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stoch. Anal. Appl. 13, 573–599 (1995)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Stoikov S., Zariphopoulou T.: Dynamic asset allocation and consumption choice in incomplete markets. Aust. Econ. Pap. 44(4), 414–454 (2005)CrossRefGoogle Scholar
  26. 26.
    Tehranchi M.: Explicit solutions of some utility maximization problems in incomplete markets. Stoch. Process. Appl. 114(1), 109–125 (2004)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Zariphopoulou T.: A solution approach to valuation with unhedgeable risks. Finance Stoch. 5(1), 61–82 (2001)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Žitković G.: A filtered version of the bipolar theorem of Brannath and Schachermayer. J. Theoret. Probab. 15, 41–61 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsETH ZurichZurichSwitzerland

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