Martingale Method for Utility Maximization

  • Jia-An Yan
Part of the Universitext book series (UTX)


The problem of the expected utility maximization for diffusion models in complete markets has been studied by many authors; see the review article of Karatzas (1989). The same problem in incomplete markets has been extensively studied by Karatzas et al. (1991). They judiciously augmented the stocks with fictitious ones to create a complete market, such that the fictitious stocks are superfluous in the optimal portfolio for the completed market. In this case, their solution is also optimal for the original incomplete market.


  1. Davis, M.H.A.: Option pricing in incomplete markets. In: Dempster, A.H., Pliska, S.R. (eds.) Mathematics of Derivative Securities, pp. 216–226. Publications of the Newton Institute, Cambridge University Press, Cambridge (1997)Google Scholar
  2. He, S.W., Wang, J.G., Yan, J.A.: Semimartingale theory and stochastic calculus. Science Press/CRC Press, Beijing/Boca Raton (1992)zbMATHGoogle Scholar
  3. Hugonnier, J., Kramkov, D., Schachermayer, W.: On utility-based pricing of contingent claims in incomplete markets. Math. Financ. 15, 203–212 (2005)MathSciNetCrossRefGoogle Scholar
  4. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Process. Springer, Berlin/Heidelberg (1987)CrossRefGoogle Scholar
  5. Karatzas, I.: Optimization problems in the theory of continuous trading. SIAM J. Control Optim. 27(6), 1221–1259 (1989)MathSciNetCrossRefGoogle Scholar
  6. Karatzas, I., Lehoczky, J.P., Shreve, S.E., Xu, G.L.: Martingale and duality methods for utility miximization in an incomplete market. SIAM J. Control Optim. 29, 702–730 (1991)MathSciNetCrossRefGoogle Scholar
  7. Lépingle, D., Mémin, J.: Sur l’intergrabilité uniforme des martingales exponentielles. Z.W. 42, 175–203 (1978)Google Scholar
  8. Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004)zbMATHGoogle Scholar
  9. Xia J.M., Yan J.A. (2000a) Martingale measure method for expected utility maximization and valuation in incomplete markets, unpublishedGoogle Scholar
  10. Xia J.M., Yan J.A. (2000b) The utility maximization approach to a martingale measure constructed via Esscher transform, unpublishedGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. and Science Press 2018

Authors and Affiliations

  • Jia-An Yan
    • 1
  1. 1.Academy of Mathematics and System ScienceChineses Academy of SciencesBeijingChina

Personalised recommendations