Skip to main content

Martingale Representations for Functionals of Lévy Processes


We describe the integrand in the martingale (or stochastic integral) representation of a square integrable functional F of a Lévy process in terms of (a derivative or difference operator acting on) a map ß F introduced in Rajeev and Fitzsimmons (Stochastics 81, 467–476, 2009). The kernels in the chaos expansion of F are also described in terms of the iterated derivative and difference operators.

This is a preview of subscription content, access via your institution.


  • Dellacherie C. and Meyer P.A. (1982). Probabilities and Potential B. North Holland.

  • Di Nunno G., Øksendal B. and Proske F. (2004). White Noise Analysis for Lévy Processes. J. Funct. Anal. 206, 109–148.

    MathSciNet  Article  MATH  Google Scholar 

  • Di Nunno G. (2007). Randon Fields: non anticipating derivatives and differentiation formulas. Infinite Dimensional Analysis and Quantum Probability 10.

  • Di Nunno G., Øksendal B. and Proske F. (2009). Malliavin Calculus for Lévy processes with Applications to Finance. Springer.

  • Émery M. (2006). Chaotic Representation Property of certain Azema martingales. Ill. J. Math. 50, 2, 395–411.

    MATH  Google Scholar 

  • He S.-W., Wang J.-G. and Yan J.-A. (1992). Semi- Martingale Theory and Stochastic Calculus. Science Press, Beijing.

    Google Scholar 

  • Itô K. (1951). Multiple Wiener integral. J. Math. Soc. Japan 3, 157–169.

    MathSciNet  Article  MATH  Google Scholar 

  • Itô K. (1956). Spectral type of the shift transformation of differential processes with stattionary. increments. Trans. Am. Math. Soc. 81, 252–263.

    Article  Google Scholar 

  • Kallenberg O. (2002). Foundations of Modern Probability. Springer.

  • Privault N. (2009). Stochastic Analysis in Discrete and Continuous Settings. Springer.

  • Nualart D. and Schoutens W. (2000). Chaotic and predictable representation for Lévy processes. Stoch. Process. Appl. 90, 109–122.

    MathSciNet  Article  MATH  Google Scholar 

  • Rajeev B. and Fitzsimmons P. (2009). A new approach to Martingale representation. Stochastics 81, 467–476.

    MathSciNet  MATH  Google Scholar 

  • Rajeev B. 2009 Stochastic Integrals and Derivatives, Bulletin of Kerala Mathematics Association. October 2009, special issue, p. 105–127.

  • Solé J.L., Utzet F. and Vives J. (2007). Chaos expansions and Malliavin Calculus for Lévy processes, Benth F. E. et al. (eds.),. Proc. 2nd Abel Symposium, Springer 1987.

  • Stroock D.W. (1987). Homogenous chaos revisited. Séminaire de Probabilités 21, 1–7.

    MathSciNet  Article  Google Scholar 

  • Wiener N. (1938). The homogeneous chaos. Amer. J. Math. 60, 897–936.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to B. Rajeev.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rajeev, B. Martingale Representations for Functionals of Lévy Processes. Sankhya A 77, 277–299 (2015).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:

Keywords and phrases

  • Martingale representation
  • Stochastic integral representation
  • Lévy processes
  • Chaos expansion
  • Stochastic derivative

AMS (2000) subject classification

  • Primary 60H10
  • 60H15
  • Secondary 60J60
  • 35K15