Advertisement

Optimal Equivalent Probability Measures under Enlarged Filtrations

  • Markus HessEmail author
Article
  • 8 Downloads

Abstract

In a general jump-diffusion Radon–Nikodym setup with stochastic Girsanov processes, we derive optimal equivalent probability measures. Optimality is measured in terms of minimum relative entropy and also by more general divergence concepts. We further provide an anticipative sufficient stochastic minimum principle and derive optimal equivalent probability measures under various enlarged filtration approaches.

Keywords

Stochastic optimization problem Stochastic maximum/minimum principle Relative entropy Radon–Nikodym density Lévy process Enlarged filtration Stochastic differential equation 

Mathematical Subject Classification

93E20 60H05 60H10 60G44 

JEL Classification

C02 C61 

Notes

References

  1. 1.
    Hess, M.: Pricing energy, weather and emission derivatives under future information. Ph.D. thesis, University Duisburg-Essen, Germany (2013) https://duepublico.uni-duisburg-essen.de/servlets/DocumentServlet?id=31060
  2. 2.
    Chan, T.: Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab 9(2), 504–528 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cont, R., Tankov, P.: Financial Modeling with Jump Processes, 1st edn. Chapman & Hall/CRC, London (2004)zbMATHGoogle Scholar
  4. 4.
    Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Finance 12(2), 99–123 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10(1), 39–52 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fujiwara, T., Miyahara, Y.: The minimal entropy martingale measures for geometric Lévy processes. Finance Stoch. 7(4), 509–531 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Goll, T., Rüschendorf, L.: Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stoch. 5, 557–581 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grandits, P., Rheinländer, T.: On the minimal entropy martingale measure. Ann. Probab. 30(3), 1003–1038 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jeanblanc, M., Klöppel, S., Miyahara, Y.: Minimal f q-martingale measures for exponential Lévy processes. Ann. Appl. Probab. 17, 1615–1638 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, Berlin (2009)CrossRefGoogle Scholar
  11. 11.
    Miyahara, Y.: [Geometric Lévy processes & MEMM] pricing model and related estimation problems. Asia-Pac. Financ. Mark. 8, 45–60 (2001)CrossRefGoogle Scholar
  12. 12.
    Rheinländer, T., Steiger, G.: The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models. Ann. Appl. Probab. 16(3), 1319–1351 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Schweizer, M.: Minimal entropy martingale measure. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 1195–1200. Wiley, Hoboken (2010)Google Scholar
  14. 14.
    Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions, 2nd edn. Springer, Berlin (2007)CrossRefGoogle Scholar
  15. 15.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005)CrossRefGoogle Scholar
  16. 16.
    Applebaum, D.: Lévy Processes and Stochastic Calculus. Second Edition, Cambridge Studies in Advanced Mathematics (2009)Google Scholar
  17. 17.
    Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Lévy Processes with Applications to Finance, 1st edn. Springer, Berlin (2009)CrossRefGoogle Scholar
  18. 18.
    Jacod, J., Protter, P.: Time reversal on Lévy processes. Ann. Probab. 16(2), 620–641 (1988)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)CrossRefGoogle Scholar
  20. 20.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, No. 68 (1999)Google Scholar
  21. 21.
    Baghery, F., Øksendal, B.: A maximum principle for stochastic control with partial information. Stoch. Anal. Appl. 25(3), 705–717 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Biagini, F., Øksendal, B.: A general stochastic calculus approach to insider trading. Appl. Math. Optim. 52, 167–181 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Di Nunno, G., Meyer-Brandis, T., Øksendal, B., Proske, F.: Optimal portfolio for an insider in a market driven by Lévy processes. Quant. Finance 6(1), 83–94 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hess, M.: An anticipative stochastic minimum principle under enlarged filtrations. SSRN working paper (2017). https://ssrn.com/abstract=3075113
  25. 25.
    Protter, P.: A Connection Between the Expansion of Filtrations and Girsanov’s Theorem. Stochastic Partial Differential Equations and Applications II, Lecture Notes in Mathematics 1390, pp. 221–224. Springer, Berlin (1989)Google Scholar
  26. 26.
    Hess, M.: Pricing temperature derivatives under weather forecasts. Int. J. Theor. Appl. Finance 21(5), 1850031 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversité Libre de BruxellesBrusselsBelgium

Personalised recommendations