Asia-Pacific Financial Markets

, Volume 16, Issue 1, pp 65–95 | Cite as

The Minimal Entropy Martingale Measures for Exponential Additive Processes

  • Tsukasa FujiwaraEmail author


In this paper, we will consider exponential additive processes as a financial market model. Under a mild condition, we will determine the minimal entropy martingale measures (MEMMs) for the exponential additive processes. To this end, we will prepare several results on the exponential moment of additive processes and integrals based on them. As an application of our result, we will deduce optimal strategy for exponential utility maximization problem. We will also investigate our result through several examples, such as time-dependent versions of double Poisson model, Merton model and Kou model.


Additive process Exponential additive process (Local) Martingale measure Minimal entropy martingale measure Exponential utility Optimal strategy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Choulli T., Stricker C. (2006) More on minimal entropy-Hellinger martingale measure. Mathematical Finance 16: 1–19CrossRefGoogle Scholar
  2. Cont R., Tankov P. (2004) Financial modelling with jump processes. Chapman & Hall/CRC Press, London/Boca Raton FLGoogle Scholar
  3. Delbaen F., Grandits P., Rheinländer T., Samperi D., Schweizer M., Stricker C. (2002) Exponential hedging and entropic penalties. Mathematical Finance 12: 99–123CrossRefGoogle Scholar
  4. Dellacherie C., Meyer P.A. (1982) Probabilities and potential B—theory of martingales. North-Holland, AmsterdamGoogle Scholar
  5. Esche F., Schweizer M. (2005) Minimal entropy preserves the Lévy property: How and why. Stochastic Processes and their Applications 115: 299–327CrossRefGoogle Scholar
  6. Fujiwara T. (2006) From the minimal entropy martingale measures to the optimal strategies for the exponential utility maximization: The case of geometric Lévy processes. Asia-Pacific Financial Markets 11: 367–391CrossRefGoogle Scholar
  7. Fujiwara T., Miyahara Y. (2003) The minimal entropy martingale measures for geometric Lévy processes. Finance and Stochastics 7: 509–531CrossRefGoogle Scholar
  8. He S.W., Wang J.G., Yan J.A. (1992) Semimartingale theory and stochastic calculus. Science Press/CRC Press, Beijing/Boca Raton, FLGoogle Scholar
  9. Henderson V., Hobson D. (2003) Coupling and option price comparisons in a jump-diffusion model. Stochastics and Stochastics Reports 75: 79–101CrossRefGoogle Scholar
  10. Hubalek F., Sgarra C. (2006) Esscher transforms and the minimal entropy martingale measure for exponential Lévy models. Quantitative Finance 6: 125–145CrossRefGoogle Scholar
  11. Ikeda N., Watanabe S. (1989) Stochastic differential equations and diffusion processes (2nd ed.). Amsterdam/Tokyo, North-Holland/KodanshaGoogle Scholar
  12. Itô K. (2004) Stochastic processes. Lectures given at Aarhus University. Springer-Verlag, BerlinGoogle Scholar
  13. Jacod J., Shiryaev A.N. (2003) Limit theorems for stochastic processes (2nd ed.). Springer, BerlinGoogle Scholar
  14. Kabanov Y., Stricker C. (2002) On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper. Mathematical Finance 12: 125–134CrossRefGoogle Scholar
  15. Kallsen J., Shiryaev A.N. (2002) The cumulant process and Esscher’s change of measure. Finance and Stochastics 6: 397–428CrossRefGoogle Scholar
  16. Kou S.G. (2002) A jump-diffusion model for option pricing. Management Science 48: 1086–1101CrossRefGoogle Scholar
  17. Merton R.C. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3: 125–144CrossRefGoogle Scholar
  18. Protter P. (2004) Stochastic integration and differential equations—a new approach Applications of Mathematics, Vol. 21 (2nd ed.). Springer, New YorkGoogle Scholar
  19. Sato K. (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, LondonGoogle Scholar
  20. Shiryaev A.N. (1999) Essentials of stochastic finance: Facts, models, theory. World Scientific Publishing, SingaporeGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of MathematicsHyogo University of Teacher EducationKato, HyogoJapan

Personalised recommendations